2. Graphing Quadratic Functions Using Standard Form
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Chapter 8
2.
Graphing Quadratic Functions Using Standard Form
| Student Learning Objectives: |
|---|
|
Why
This lesson delves into the intricacies of graphing quadratic functions, particularly when they are presented in standard form. It emphasizes the importance of understanding the vertex, axis of symmetry, and other key characteristics of a parabola. Through various examples, the lesson illustrates how to determine whether a quadratic function is expressed in standard form and how to graph it accordingly. The use cases highlighted include real-world scenarios like modeling the trajectory of a kicked ball and understanding the height of fireworks after launch. These practical applications underscore the relevance of mastering quadratic functions in everyday life.
Lesson Settings & Tools
| | 13 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Graphing Quadratic Functions Using Standard Form
Slide of 13
This lesson explains what the standard form of a quadratic function is and how to draw its graph. How to determine various characteristics of a quadratic function such as the axis of symmetry, the vertex, and the y-intercept will then be understood.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Expanding the Vertex Form of a Quadratic Function
Consider the vertex form of a quadratic function y = a(x-p)^2+q and its corresponding parabola. In the applet, adjust the parameters of each parabola's vertex. Investigate how the expanded form of the equation is rewritten according to the new parameters.
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Expanding the Intercept Form of a Quadratic Function
Consider the intercept form of a quadratic function and its corresponding parabola. In the applet, adjust the parameters of each parabola's x-intercepts. Investigate how the expanded form of the equation is rewritten according to the new parameters.
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Discussion
Standard Form of a Quadratic Function
Besides the intercept and the vertex forms, another essential and common form of a quadratic function is its standard form.
The standard form of a quadratic function is a quadratic function written in a specific format.
f(x)=ax^2+bx+c
Here, a, b, and c are real numbers with a≠ 0. The term with the highest degree — the quadratic term — is written first, then the linear term, followed by the constant term. The standard form of the function can be used to determine the direction of the parabola, the y-intercept, the axis of symmetry, and the vertex.
| Direction of the Graph | Opens upward when a> 0 |
|---|---|
| Opens downward when a< 0 | |
| y-intercept | c |
| Axis of Symmetry | x = -b/2a |
| Vertex | (-b/2a,f(-b/2a)) |
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Is the Quadratic Function Written in Standard Form?
Determine whether the given quadratic function is expressed in standard form.
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Discussion
Graphing a Quadratic Function in Standard Form
Given a quadratic function in standard form, some characteristics of its corresponding parabola can be determined. Consider an example quadratic function. f(x) = x^2-4x+3 To draw the graph of the function written in standard form, there are five steps to follow.
1
Identify and Graph the Axis of Symmetry
expand_more The axis of symmetry can be found by determining a and b in the form ax^2 + bx + c. f(x) = x^2-4x+3 ⇕ f(x) = 1x^2+( -4)x+ 3
In the given function, a and b are 1 and -4, respectively. Now, these values will be substituted into the formula for the equation of the axis of symmetry.
x=-b/2a
SubstituteII
a= 1, b= -4
x=--4/2( 1)
▼
Evaluate right-hand side
IdPropMult
Identity Property of Multiplication
x=--4/2
RemoveNegFracAndNum
- - a/b= a/b
x=4/2
CalcQuot
Calculate quotient
x=2
The axis of symmetry of the function is the vertical line with equation x=2.
2
Determine and Plot the Vertex
expand_more The axis of symmetry intersects the parabola at its vertex. Therefore, x=2 is the x-coordinate of the vertex. To find the y-coordinate of the vertex, x=2 can be substituted into the function rule.
The vertex of the parabola is at (2,-1).
3
Determine and Plot the y-intercept
expand_more The y-intercept can be determined by using the constant term c of the given function. In this case, c is equal to 3. This means that the y-intercept occurs at (0,3).
4
Reflect the y-intercept Across the Axis of Symmetry
expand_more The axis of symmetry divides the graph into two mirror images. Therefore, the reflection of the y-intercept across the axis of symmetry is also on the parabola.
If the vertex of a quadratic function lies on the y-axis, then any point that lies on the graph other than the vertex should be found and reflected across the axis of symmetry. In this case, the y-axis is the axis of symmetry.
5
Draw the Parabola
expand_more Finally, connect the points with a smooth curve to graph the parabola.
The graph of the function opens upward. This is expected, since the value of a is 1, which is a positive number.
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Determining the Vertex of a Quadratic Function Written in Standard Form
Find the coordinates of the vertex of the parabola that corresponds to the quadratic function written in standard form. If necessary, round the answer to 2 decimal places.
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Example
Will a German Shepherd Jump Over the Fence?
Zain lives in a quiet rural town where it seems not a lot happens. He decides to adopt a German shepherd puppy. After a few short weeks the puppy is growing so fast, they are afraid that soon the pup will be able to jump over the fence around their garden!
External credits: @brgfx
Zain surfs the net and finds that a researcher has actually modeled the path of an adult German shepherd's jump! The path Zain found is a parabolic path modeled by a quadratic function. h(x) = -0.075 x^2 + 1.35x Here, x is the dog's horizontal distance from the jump spot and h(x) is the height of that jump. Both values are given in feet.
a Draw the function's graph.
b If the height of the fence is 7 feet, will their dog, when it is an adult, manage to jump over the fence?
c Zosia and Zain made one more observation about their puppy's mad hops. The puppy can already jump, horizontally, over a 5-foot-wide hole. How much further will the dog be able to jump as an adult?
Answer
a
b No
c 13 feet
Hint
a Start by determining the axis of symmetry and the vertex of the parabola. Then, find two more points that lie on the curve.
b Consider the coordinates of the parabola's vertex.
c Use the graph from Part A.
Solution
a Consider the given quadratic function.
h(x) = -0.075 x^2 + 1.35x This function is written in standard form. To graph a quadratic function written in this form, there are five steps to follow.
- Identify and graph the axis of symmetry
- Determine and plot the vertex
- Determine and plot the y-intercept
- Reflect the y-intercept in the axis of symmetry
- Draw the parabola
These steps will be done one at a time.
The Axis of Symmetry
To find the equation of the axis of symmetry, the coefficients a, b, and c should first be found. Consider the given function. h(x) = -0.075 x^2 + 1.35x ⇕ h(x) = -0.075 x^2 + 1.35x + 0 Here, a = -0.075, b = 1.35, and c= 0. The axis of symmetry is a vertical line with equation x = - b2a.
x = - b/2a
SubstituteII
a= -0.075, b= 1.35
x = - 1.35/2( -0.075)
▼
Evaluate right-hand side
MultPosNeg
a(- b)=- a * b
x = - 1.35/-0.15
DivNegNeg
- a/- b=a/b
x = 1.35/0.15
CalcQuot
Calculate quotient
x = 9
The axis of symmetry is the vertical line x=9.
The Vertex
The vertex of the function lies on the axis of symmetry. This means that the vertex's x-coordinate is 9. Now, to find the y-coordinate, x=9 will be substituted into the function rule.
h(x) = -0.075 x^2 + 1.35x
Substitute
x= 9
h( 9) = -0.075 ( 9)^2 + 1.35( 9)
▼
Evaluate right-hand side
CalcPowProd
Calculate power and product
h(9) = -0.075 (81) + 12.15
MultNegPos
(- a)b = - ab
h(9) = - 6.075 + 12.15
AddTerms
Add terms
h(9) = 6.075
The vertex of the function is at (9, 6.075).
The y-intercept
The y-intercept is given by the constant term c of the function rule. In this case this term is equal to 0, which means that the y-intercept occurs at (0,0) — the origin.
Reflecting the y-intercept
Now, another point that lies on the parabola can be found by reflecting the y-intercept in the axis of symmetry.
The third point that lies on the parabola is at (18, 0).
Drawing the Parabola
Finally, the points will be connected with a smooth curve to graph the parabolic shape. Since the function represents a dog's jump, negative values of the function will not be included.
b To determine whether the adult dog will be able to jump over the fence, the fence will be drawn in the same coordinate plane. Assume that the fence is placed at the axis of symmetry, where the height of the jump would the greatest.
Since the greatest height that the dog will reach is 6.075 feet and the height of the fence is 7 feet, the dog will not be able to jump over the fence.
c It is known that the dog can jump over a hole whose width is 5 feet. Consider the graph of the function once again and note the horizontal distance that the dog will be able to jump as an adult.
The maximum horizontal distance that the dog will be able to jump corresponds to the distance between the x-intercepts. This distance is 18 feet. To find how much further the dog will be able to jump, the difference between 18 and 5 must be calculated. 18 - 5 = 13feet What a pup!
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Discussion
Rewriting a Quadratic Function in Standard Form
Depending on the conditions, it is convenient to rewrite a quadratic function given in intercept or vertex form in its standard form.
Both the vertex and intercept forms of a quadratic function can always be rewritten in standard form.
| Form | Equation | How to Rewrite? |
|---|---|---|
| Vertex Form | y = a(x-h)^2+k | Expand (x-h)^2, distribute a, and combine like terms. |
| Intercept Form (also called Factored Form) |
y = a(x-p)(x-q) | Multiply a(x-p)(x-q) and combine like terms. |
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Example
Graphing Quadratic Functions to Represent Profit
The mayor of Zain's sleepy town wants to do something exciting for the community. Lucky her, she noticed an increasing interest in people taking up skateboarding, so she knew that building a new skate ramp would be a huge hit.
She was right! Now a local skateboard company named SuTeKi Sk8 sees this as an opportunity to make some big sales. SuTeKi Sk8 investigates the profit of its two coolest models, Sketchtastic and Sugoi Sketch. They collect data and find a relationship between the price and the profit on sales of both skateboards. The following quadratic functions represent their results.
| Skateboard | Relationship |
|---|---|
| Sketchtastic | y = -0.015(x-50)^2 + 37 |
| Sugoi Sketch | y = -0.008(x-65)^2+28 |
In these function rules, x is the price in dollars and y is the corresponding profit in thousands of dollars.
a Rewrite the quadratic functions in standard form and graph them in the same coordinate plane.
c Using the graph, estimate the profit on sales if they set the price at $80 for each model. Then, compare the results with the exact values.
Answer
a Standard Forms:
| Skateboard | Vertex Form | Standard Form |
|---|---|---|
| Sketchtastic | y = -0.015(x-50)^2 + 37 | y = -0.015x^2+1.5x-0.5 |
| Sugoi Sketch | y = -0.008(x-65)^2+28 | y = -0.008x^2+1.04x-5.8 |
Graphs:
b See solution.
c Estimation:
| Skateboard | Profit on Sales |
|---|---|
| Sketchtastic | ≈ $23 000 |
| Sugoi Sketch | ≈ $26 000 |
Comparison: See solution.
Hint
a To rewrite the quadratic functions in standard form, start by expanding the squared expression. To draw a parabola, determine the axis of symmetry and the vertex. Then, find two more points on the curve.
b The vertex of a parabola represents the maximum or minimum value of the function.
c To estimate the profit, look at the values of the function at x=80. Then, substitute x=80 into the function rules to calculate the exact values.
Solution
a The quadratic functions that represent the data about the skateboard models are given in vertex form.
| Vertex Form: y = a(x- p)^2 + q | |
|---|---|
| Sketchtastic | Sugoi Sketch |
| y = -0.015(x- 50)^2 + 37 | y = -0.008(x- 65)^2+ 28 |
The given functions can be rewritten in standard form by expanding the squared expression. First, the function corresponding to the Sketchtastic model will be rewritten.
y = -0.015(x-50)^2+37
▼
Rewrite
ExpandNegPerfectSquare
(a-b)^2=a^2-2ab+b^2
y = -0.015( x^2 - 2x(50) + 50^2)+37
CalcPowProd
Calculate power and product
y = -0.015( x^2 - 100x + 2500)+37
Distr
Distribute -0.015
y = -0.015x^2 + 1.5x - 37.5+37
AddTerms
Add terms
y = -0.015x^2+1.5x-0.5
By following the same procedure, the second function that represents Sugoi Sketch can also be rewritten in standard form.
| Sketchtastic | Sugoi Sketch | |
|---|---|---|
| Given Function | y = -0.015(x-50)^2+37 | y = -0.008(x-65)^2+28 |
| Standard Form | y = -0.015x^2+1.5x-0.5 | y = -0.008x^2+1.04x-5.8 |
The functions can now be graphed using the standard form. To do so, there are five steps to follow.
- Identify and graph the axis of symmetry.
- Determine and plot the vertex.
- Determine and plot the y-intercept.
- Reflect the y-intercept in the axis of symmetry.
- Draw the parabola.
The axis of symmetry is given by the equation x = - b2a, where a is the coefficient of the x^2-term and b is the linear coefficient. Use a calculator to evaluate the equations if necessary.
| Axis of Symmetry: x=- b/2a | |
|---|---|
| y = -0.015x^2+1.5x-0.5 | y = -0.008x^2+1.04x-5.8 |
| x = -1.5/2(-0.015) | x = -1.04/2(-0.008) |
| x= 50 | x= 65 |
The axes of symmetry of the parabolas are the vertical lines with equations x=50 and x=65.
The vertices of the parabolas lie on the corresponding axes of symmetry. This means that x=50 and x=65 are the x-coordinates of the vertices. Now, these values will be substituted for x into the function rules to find the y-coordinates.
| y = -0.015x^2+1.5x-0.5 | y = -0.008x^2+1.04x-5.8 | |
|---|---|---|
| x-coordinate | 50 | 65 |
| y-coordinate | y=-0.015( 50)^2+1.5( 50)-0.5 ⇕ y = 37 |
y=-0.008( 65)^2+1.04( 65)-5.8 ⇕ y = 28 |
| Vertex | ( 50, 37) | ( 65, 28) |
The vertices can now be added to the graph.
The y-intercepts are given by the constant term of the function rules.
| Function Rule | y-intercept |
|---|---|
| y = -0.015x^2+1.5x -0.5 | -0.5 |
| y = -0.008x^2+1.04x -5.8 | -5.8 |
Next, another point that lies on each parabola can be found by reflecting the y-intercept in the axis of symmetry.
Finally, the corresponding points will be connected with a smooth curve to graph the parabolic shapes.
b In the given functions, x is the price in dollars and y is the corresponding the profit on sales in thousands of dollars. Both functions reach their maximum values at their corresponding vertices.
The x-coordinate of each vertex is the price of the corresponding skateboard at which the profit of the company is maximized. This means that the vertex represents the optimal price and its corresponding profit. The company should use these values to make the highest return on sales.
c To estimate the profit of the skateboards if the price is set at $80, consider the graph of the functions once again. Since that price is given, a line corresponding to the price of $80 will be added to the graph.
From the graph, it can be seen that the profits of skateboards Sketchtastic and Sugoi Sketch are about 23 and 26 dollars in the thousands, respectively. To verify this, x= 80 will be substituted in both function rules.
| Sketchtastic | Sugoi Sketch | |
|---|---|---|
| x | 80 | 80 |
| Substitute | -0.015( 80)^2+1.5( 80)-0.5 | -0.008( 80)^2+1.04( 80)-5.8 |
| Evaluate | 23.5 | 26.2 |
The estimated values are slightly less than the exact values.
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Example
Using a Quadratic Function to Model the Trajectory of a Kicked Ball
Zain's sleepy town is becoming more exciting these days. Two rival soccer teams, feeling the town's energy, meet at the park. The two captains begin a duel of sorts, and a soccer ball is kicked from the ground level towards a goal 80 feet away. The ball follows a parabolic path and is caught at the height of 4 78 feet by a goalkeeper. The goalkeeper is standing 15 feet in front of the goal.
External credits: Jeffrey F Lin
a In its intercept form, write the equation of the quadratic function that represents the path of the ball, assuming that it would hit the goal's line.
b Rewrite the obtained function in standard form. Then, graph the function.
c What is the maximum height of the ball?
d Is the ball already falling when its horizontal distance from the kicker is 50 feet?
Answer
a y=-0.005x(x-80)
b Standard Form: y = -0.005x^2+0.4x
Graph:
c 8 feet
d Yes, see solution.
Hint
a Illustrate the given situation on a coordinate plane. Mark the starting position of the ball and the goal's line on the x-axis. Then, use intercept form of a quadratic function to find the function rule.
b Rewrite the function obtained in Part A by expanding the function rule. Next, determine the axis of symmetry and the vertex of the given function. Then, find two more points that lie on the parabola.
c At what point does a parabola reach its maximum or minimum value?
d Consider the graph of the function. For what values of x is the function decreasing?
Solution
a To write the quadratic function that represents the path of the ball, consider all the information given.
- A player kicks the ball from a distance of 80 feet.
- If the goalkeeper does not catch the ball, it will hit the goal's line.
- The goalkeeper — standing 15 feet in front of the goal — catches the ball at a height of 4 78 feet.
The given situation can be illustrated on a coordinate plane. Assume that the starting position of the ball is at the origin. The x-axis represents the horizontal distance from the starting point and the y-axis represents the height of the ball.
There are two x-intercepts. These are the starting point at (0,0) and the goal's line at (80,0). Therefore, to write the quadratic function representing the path of the ball, the intercept form can be used. The zeros of the parabola are 0 and 80.
y = a(x-p)(x-q)
SubstituteII
p= 0, q= 80
y = a(x- 0)(x- 80)
IdPropAdd
Identity Property of Addition
y = ax(x-80)
In the given scenario, the ball does not hit the goal's line. Instead, the goalkeeper, standing 15 feet in front of the goal, catches the ball at a height of 4 78 feet.
When the goalkeeper catches the ball, its horizontal distance from the starting point is 65 feet. This means that the value of the function at x= 65 is y = 4 78. These values can be substituted into the intercept form to find the missing coefficient a.
y = ax(x-80)
SubstituteII
x= 65, y= 4 78
4 78 = a( 65)( 65-80)
▼
Solve for a
SubTerms
Subtract terms
4 78 = a(65)(-15)
MultPosNeg
a(- b)=- a * b
4 78 = a(-975)
MixedToFrac
a bc=a* c+b/c
39/8 = a(-975)
DivEqn
.LHS /(-975).=.RHS /(-975).
39/8/-975 = a
MoveNegDenomToFrac
Put minus sign in front of fraction
-39/8/975 = a
DivFracD
a/c/b= a/b* c
-39/7800 = a
UseCalc
Use a calculator
-0.005 = a
RearrangeEqn
Rearrange equation
a = -0.005
Finally, the quadratic function that represents the ball's path can be written. y = ax(x-80) ⇓ y = -0.005x(x-80)
b In Part A, a quadratic function written in intercept form has been obtained.
y = -0.005x(x-80) To rewrite this function in standard form, the expression -0.005x will be distributed. y = -0.005x(x-80) ⇕ y = -0.005x^2+0.4x The function can now be graphed using the standard form. To do so, there are five steps to follow.
- Identify and graph the axis of symmetry
- Determine and plot the vertex
- Determine and plot the y-intercept
- Reflect the y-intercept in the axis of symmetry
- Draw the parabola
The axis of symmetry is the vertical line with equation x = - b2a, where a is the coefficient of the x^2-term and b is the linear coefficient.
The equation of the axis of symmetry is x=40.
The parabola's vertex lies on the axis of symmetry. This means that x=40 is the x-coordinate of the vertex. Now, this value will be substituted into the function rule to find the y-coordinate.
The vertex of the parabola is at (40, 8).
The y-intercept is given by the constant term of the function rule, which in this case is 0. Therefore, the y-intercept is at the origin (0,0).
Now, another point that lies on the parabola can be found by reflecting the y-intercept in the axis of symmetry.
The third point lies at (80, 0). Finally, the three points will be connected with a smooth curve to graph the parabolic shape. Since the function represents height of the ball, negative values of the function will not be considered.
c To determine the maximum height of the ball, the maximum of the given function will be considered.
A parabola that opens downward reaches its maximum at the vertex. In Part B, it was obtained that the vertex of the given function is at (40, 8). Therefore, the maximum height of the ball is 8 feet.
d Consider the graph of the given function once again. The horizontal distance d = 50 from the kicker — who is placed at the origin — will be marked in the graph.
Recall that the x-coordinate of the vertex is 40. Since the given distance is greater than 40, it can be said that the ball is already falling at x=50.
If the goalie understands this math concept, and can apply it in the heat of a soccer match, they are surely going to be an amazing player and maybe some day a sports scientist. What exciting times in what Zain thought was a sleepy town!
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Rewriting a Quadratic Function in Standard Form
The quadratic functions are given in either intercept or vertex form. State the coefficients a, b, and c of the standard form y = ax^2+bx+c.
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Closure
Graphing Quadratic Functions in Different Forms
The three most common forms of a quadratic function are standard form, intercept form, and vertex form.
| Form | Formula |
|---|---|
| Standard | y=ax^2+bx+c |
| Intercept | y=a(x-p)(x-q) |
| Vertex | y=a(x-h)^2+k |
In these forms, a is not equal to 0. Sometimes, the function rule of a quadratic function is not known. Instead of the function rule, the y-intercept and at least two more points that lie on the parabola may be given. In this case, the function can also be graphed using its standard form. The coefficients of the standard form can be found by following four steps.
- Use the y-intercept to find the constant term c
- Substitute the coordinates of the points into the general standard form for x and y
- Solve the obtained system of equations for a and b
- Write the standard form of the function
An example will be considered.
Example
Consider the y-intercept of a parabola and two of its points. y-intercept:& (0, 1) Points:& (1, 1), (2, 5) Now, the standard form of the function y=ax^2+bx+c will be found by following the mentioned steps.
1
Use the y-intercept to Find c
expand_more The y-intercept occurs at (0, 1). This means that c is equal to 1.
y=ax^2 + bx + c ⇓ y=ax^2 + bx + 1
2
Substitute the Coordinates of the Points
expand_more Next, the coordinates of the given points will be substituted for x and y into the obtained partial equation. Each point will be considered one at a time starting with (1,1).
y = ax^2 + bx + 1
SubstituteII
x= 1, y= 1
1 = a( 1^2) + b( 1) + 1
▼
Simplify
BaseOne
1^a=1
1 = a(1)+b(1)+1
IdPropMult
Identity Property of Multiplication
1 = a+b+1
SubEqn
LHS-1=RHS-1
0 = a+b
RearrangeEqn
Rearrange equation
a+b = 0
Similarly, the coordinates of the second point (2,5) will be substituted.
y = ax^2 + bx + 1
SubstituteII
x= 2, y= 5
5 = a( 2^2) + b( 2) + 1
▼
Simplify
CalcPow
Calculate power
5 = a(4)+b(2)+1
CommutativePropMult
Commutative Property of Multiplication
5 = 4a+2b+1
SubEqn
LHS-1=RHS-1
4 = 4a+2b
RearrangeEqn
Rearrange equation
4a+2b = 4
The obtained equations are a+b=0 and 4a+2b=4.
3
Solve the System of Equations
expand_more Now, consider the system of equations formed by the obtained equations.
a+b=0 & (I) 4a+2b=4 & (II)
The system can be solved by using the Substitution Method. For simplicity, in this case a-variable will be isolated in Equation (I).
Now, Equation (II) will be solved for b.
a=- b 4(- b)+2b=4
▼
(II): Solve for b
MultPosNeg
(II): a(- b)=- a * b
a=- b -4b+2b=4
AddTerms
(II): Add terms
a=- b -2b=4
DivEqn
(II): .LHS /(-2).=.RHS /(-2).
a=- b b=4/-2
MoveNegDenomToFrac
(II): Put minus sign in front of fraction
a=- b b=-4/2
CalcQuot
Calculate quotient
a=- b b=-2
Substitute
(I): b= -2
a=- ( -2) b=-2
NegNeg
(I): - (- a)=a
a= 2 b= -2
4
Write the Standard Form
expand_more Finally, the standard form can be written.
y = 2x^2 + ( -2)x + 1 ⇕ y = 2x^2-2x+1
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Graphing Quadratic Functions Using Standard Form
Exercise 2.1
In this function, x and y are given in feet.
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Notice that the point where the cables are anchored is the lowest point on the parabola, which is the vertex. Before finding the x-coordinate of the vertex, let's identify the values of a, b, and c for the given quadratic function written in standard form. y=1/700x^2-x+235 ⇕ y= 1/700x^2+( - 1)x+ 235 We see that a= 1700, b= - 1, and c= 235. Let's now substitute a= 1700, and b= - 1 into the formula x=- b2 a to find the x-coordinate of the vertex.
If we place the first tower at x=0, the point where the cables are anchored is 350 feet far from each tower.
We know that the cable is at the road level meaning the road and the lowest point of the cable have the same height. The y-coordinate of the vertex will give us how high the lowest point of the cable is. To find it, we will substitute the x-coordinate of the vertex we found in Part A into the given function rule.
Therefore, the road level is 60 feet above the water.
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Hint & Answer
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Solution
Assuming that y=0 is the road level, the parabola that describes the span can be modeled by the following function. y = ax^2 + b In this function rule, a and b are unknown constants.
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We know that beams reach the height of 85 meters above the road midway between the towers at x=0. This means that the value of the quadratic function that describes the span is 85 for x = 0. Substitute these values into the function rule y = ax^2 + b and solve for b.
The constant b is 85. Since b is the constant term in the equation, you can also interpret it as the y-intercept of the parabola.
Start by determining the missing constant a. In Part A, we already found that b=85, which gives the following function rule. y = ax^2 + 85 To determine a, we need another point that lies on the parabola and where the span intersects the road. Since the bridge is 400 meters long, the middle point must lie 200 meters from each point of intersection. Therefore, the span of the bridge intersects the road at the points (-200, 0) and (200, 0).
Let's now substitute x = 200 and y = 0 into the function rule and solve for a.
We can now write the function that describes the span. y = - 0.002125 x^2 + 85 Finally, we will determine the distance between the towers. The span meets the water 49 meters below the bridge. This corresponds to y=-49, because y=0 is the road level. If we determine the x-coordinates of the points that lie on the parabola when y=-49, we will be able calculate the distance between the points.
To find x_1 and x_2, we will substitute y=- 49 into the function rule and solve for x.
Therefore, x_1 is about -251 and x_2 is about 251. Since both points lie on the water level, the difference between x_2 and x_1 will give us the distance between towers.
The distance between the towers is approximately 502 meters.
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The function R(n) — where n is the number of $5 price decreases — represents the revenue from board games sales. R(n) = (80 - 5n)(120 + 10n) We want to find an optimal n-value that maximizes the revenue. To do so, we need to find the x-coordinate of the vertex which is the maximum point on the corresponding parabola. Before that, let's rewrite the function in standard form.
We can now identify the values of a, b, and c in the standard form ax^2+bx+c. R(n)=-50n^2 + 200n + 9600 ⇕ y= - 50n^2+ 200n+ 9600 Notice that that a= - 50, b= 200, and c= 9600. We can now find the vertex by stating the x- and y-coordinate in terms of a and b. Vertex ( - b/2 a, R( - b/2 a ) ) Let's find the x-coordinate of the vertex.
This means that two $5 decreases in price gives the greatest revenue. Let's calculate the optimal price in this case. $80 - 2* $5 = $ 70 Therefore, the manufacturer should charge $ 70 for the board game.
We will start by writing a function that represents the revenue when the manufacturer expects to sell 15 more board games for each $ 8 decrease in price. R(n)=(unit price)(units sold) ⇓ R(n)=(80- 8n)(120+ 15n) Similar to Part A, the x-coordinate of vertex can be found by rewriting the obtained function rule in standard form and using the formula - b2a. This gives that the x-coordinate of the vertex is 1. By substituting this number into the function rule, we will obtain the maximum value of the function R(n).
In this model, the estimated maximum revenue is $9720. Similarly, we can calculate the maximum value of the given function R(n).
| Function | - b/2a | Substitute | R(- b/2a) |
|---|---|---|---|
| R(n) | 1 | - | R( 1)=9720 |
| R(n) | 2 | R(n) = (80 - 5( 2))(120 + 10( 2)) | R( 2)=9800 |
We see that the revenue model R results in a greater maximum monthly revenue. Therefore, the answer is no.
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Graphing Quadratic Functions Using Standard Form
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