Graphing Quadratic Functions Using Standard Form

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

Calculate quotient

x=2

The axis of symmetry of the function is the vertical line with equation x=2.

Determine and Plot the Vertex

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.

f(x)=x^2-4x+3

x= 2

f( 2)= 2^2-4( 2)+3

▼

Evaluate right-hand side

CalcPowProd

Calculate power and product

f(2)=4-8+3

AddSubTerms

Add and subtract terms

f(2)=-1

The vertex of the parabola is at (2,-1).

Determine and Plot the y-intercept

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).

Reflect the y-intercept Across the Axis of Symmetry

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.

Draw the Parabola

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.

Pop Quiz

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.

Interactive applet asking to find the vertex of a quadratic function written in standard form

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!

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

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

a= -0.075, b= 1.35

x = - 1.35/2( -0.075)

▼

Evaluate right-hand side

a(- b)=- a * b

x = - 1.35/-0.15

DivNegNeg

- a/- b=a/b

x = 1.35/0.15

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

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

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!

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.

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.

b What does the vertex of each parabola represent in this context?

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

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.

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)

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)

x= 65, y= 4 78

4 78 = a( 65)( 65-80)

▼

Solve for a

SubTerms

Subtract terms

4 78 = a(65)(-15)

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.

x = - b/2a

b= 0.4, a= -0.005

x = - 0.4/2( -0.005)

▼

Evaluate right-hand side

a(- b)=- a * b

x = - 0.4/-0.01

DivNegNeg

- a/- b=a/b

x = 0.4/0.01

Calculate quotient

x = 40

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.

y = -0.005x^2+0.4x

x= 40

y = -0.005( 40)^2+0.4( 40)

▼

Evaluate right-hand side

CalcPow

Calculate power

y = -0.005(1600) + 0.4(40)

Multiply

y = -8 + 16

Add terms

y = 8

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!

Pop Quiz

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.

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.

Use the y-intercept to Find c

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

Substitute the Coordinates of the Points

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

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

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.

Solve the System of Equations

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).

a+b=0 4a+2b=4

SubEqn

(I):LHS-b=RHS-b

a=- b 4a+2b=4

(II):a= - b

a=- b 4( - b)+2b=4

Now, Equation (II) will be solved for b.

a=- b 4(- b)+2b=4

▼

(II): Solve for b

(II): a(- b)=- a * b

a=- b -4b+2b=4

(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

Calculate quotient

a=- b b=-2