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| 13 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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)=ax2+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=-2ab |
Vertex | (-2ab,f(-2ab)) |
Determine whether the given quadratic function is expressed in standard form.
a=1, b=-4
Identity Property of Multiplication
-b-a=ba
Calculate quotient
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).
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.
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.
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.
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!
These steps will be done one at a time.
x=9
Calculate power and product
(-a)b=-ab
Add terms
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.
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).
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.
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. |
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.
Skateboard | Vertex Form | Standard Form |
---|---|---|
Sketchtastic | y=-0.015(x−50)2+37 | y=-0.015x2+1.5x−0.5 |
Sugoi Sketch | y=-0.008(x−65)2+28 | y=-0.008x2+1.04x−5.8 |
Graphs:
Skateboard | Profit on Sales |
---|---|
Sketchtastic | ≈$23000 |
Sugoi Sketch | ≈$26000 |
Comparison: See solution.
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 |
(a−b)2=a2−2ab+b2
Calculate power and product
Distribute -0.015
Add terms
Sketchtastic | Sugoi Sketch | |
---|---|---|
Given Function | y=-0.015(x−50)2+37 | y=-0.008(x−65)2+28 |
Standard Form | y=-0.015x2+1.5x−0.5 | y=-0.008x2+1.04x−5.8 |
The functions can now be graphed using the standard form. To do so, there are five steps to follow.
The axis of symmetry is given by the equation x=2a-b, where a is the coefficient of the x2-term and b is the linear coefficient. Use a calculator to evaluate the equations if necessary.
Axis of Symmetry: x=2a-b | |
---|---|
y=-0.015x2+1.5x−0.5 | y=-0.008x2+1.04x−5.8 |
x=2(-0.015)-1.5 | x=2(-0.008)-1.04 |
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=-0.015x2+1.5x−0.5 | y=-0.008x2+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=-0.015x2+1.5x−0.5 | -0.5 |
y=-0.008x2+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.
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.
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.
Graph:
x=65, y=487
Subtract terms
a(-b)=-a⋅b
acb=ca⋅c+b
LHS/(-975)=RHS/(-975)
Put minus sign in front of fraction
ba/c=b⋅ca
Use a calculator
Rearrange equation
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.
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.
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!
The quadratic functions are given in either intercept or vertex form. State the coefficients a, b, and c of the standard form y=ax2+bx+c.
The three most common forms of a quadratic function are standard form, intercept form, and vertex form.
Form | Formula |
---|---|
Standard | y=ax2+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.
An example will be considered.
x=1, y=1
1a=1
Identity Property of Multiplication
LHS−1=RHS−1
Rearrange equation
x=2, y=5
Calculate power
Commutative Property of Multiplication
LHS−1=RHS−1
Rearrange equation
(II): a(-b)=-a⋅b
(II): Add terms
(II): LHS/(-2)=RHS/(-2)
(II): Put minus sign in front of fraction
Calculate quotient
(I): b=-2
(I): -(-a)=a
Consider the following parabola.
The vertex of the parabola is the highest or lowest point on the curve.
The parabola opens upward and therefore the vertex is its minimum point. We see the vertex is the point (2,-2).
Now we will find the equation of the axis of symmetry. The axis of symmetry is the vertical line passing through the vertex, and it divides the parabola into two mirrored images.
The equation of the axis of symmetry is x=2.
Finally, we can find the y-intercept. Recall that the y-intercept is the point where the graph intercepts the y-axis.
As we can see, the y-intercept is located at (0,1).
Consider each of the quadratic functions.
We have a quadratic function written in standard form. f(x)= ax^2+ bx+ c This form can give us a lot of information about the parabola by observing the values of a, b, and c. f(x)=4x^2-16x ⇕ f(x) = 4x^2 + ( -16)x + 0 We see that for the given function a= 4, b= -16, and c= 0. Let's consider the point at which the curve of the parabola changes direction.
This point is the vertex of the parabola. If we want to calculate the x-coordinate of this point, we can substitute the given values of a and b into the expression - b2a and simplify.
Let's now substitute this value into the given function rule to find the y-coordinate of the vertex.
Therefore, the vertex is (2,- 16).
Similar to Part A, consider the given function written in standard form. g(x)=-6x^2+3x ⇕ g(x) = -6x^2 + 3x + 0 We see that for the given equation a= -6, b= 3, and c= 0. The x-coordinate of the vertex of the parabola defines the axis of symmetry. Once again, let's substitute the values of a and b into the expression - b2a and simplify.
Remember that the axis of symmetry is the vertical line through the vertex, and divides a parabola into two mirror images. Since every point on this line will have the same x-coordinate as the vertex, we can write its equation. x=1/4 ⇔ x = 0.25
Just as in Parts A and B, determine the values of a, b, and c in the given standard form of the quadratic function. y=- x^2+6x-2 ⇕ y = -1x^2 + 6x + ( -2) We see that for the given equation a= -1, b= 6, and c= -2. Now, let's find the x-coordinate of the vertex by substituting these values into the expression - b2a and simplify.
The x-coordinate of the vertex is 3. This means that we can write the equation of the axis of symmetry. x=3 Finally, let's substitute x=3 into the given function to find the y-coordinate of the vertex.
The vertex is (3,7). This, along the axis of symmetry we found, corresponds to option D.
Consider each of the given quadratic functions.
We have a quadratic function written in standard form. y= ax^2+ bx+ c This equation can give us a lot of information about the parabola by observing the values of a, b, and c. Consider the given quadratic function. y = 3x^2 - 12x + 12 ⇕ y = 3x^2+( -12)x + 12 We see that for the given function rule a= 3, b= -12, and c= 12. The point at which the graph of a parabola changes direction also defines the maximum or minimum point of the graph. Whether the parabola has a minimum or maximum is determined by the value of a.
Since the given value of a is positive, the parabola reaches a minimum value at the vertex.
Similar to Part A, we are given a quadratic function written in standard form.
y = -4x^2 - 4x + 4
⇕
y = -4x^2+( -4)x + 4
We see that for the given function rule a= -4, b= -4, and c= 4. It is already given that the function reaches a maximum value. To find this value, we first need to determine the vertex of the function.
We can find the x-coordinate of the vertex by substituting the values of a and b into the expression - b2a and simplifying.
To find the y-coordinate of the vertex, think of y as a function of x. By substituting the x-coordinate of the vertex into the given equation and simplifying, we will get the y-coordinate of the vertex.
This means that y=5 is the maximum value of the given function.
Just as in Part A and Part B, consider the given quadratic function.
y = -2x^2 + 4x + 1
We see that for the given function rule a= -2, b= 4, and c= 1.
Consider the point at which the curve of the parabola changes direction.
This point is the vertex of the parabola. Once again, if we want to calculate the x-coordinate of this point, we can substitute the given values of a and b into the expression - b2a and simplify.
The point at which the graph of a parabola changes direction also defines the maximum or minimum point of the graph. Whether the parabola has a minimum or maximum is determined by the value of a.
Since the given value of a is negative, the parabola has a maximum value at the vertex. To find this value, think of y as a function of x. By substituting the x-coordinate of the vertex into the given equation and simplifying, we will obtain the y-coordinate of the vertex.
Therefore, the maximum value of the function is 3.
A fireworks stand in the local neighborhood introduced a new firework model that reaches great heights after it is launched.
We have a quadratic function written in standard form. h= at^2 + bt + c This equation can give us information about the parabola by observing the values of a, b, and c. h = -32t^2+384t ⇕ h = -32t^2+ 384t+ 0 We see that for the given equation a= -32, b= 384, and c= 0. Since -16<0, the parabola opens downward and the vertex is a maximum point. Therefore, the x-coordinate of the vertex will give us the time when the firework explodes. To calculate it, we can substitute the given values of a and b into the expression - b2a and simplify.
Therefore, the firework will explode 6 seconds after we launch it.
In Part A we have found that the firework reaches its maximum height for t=6. Also, at this time is when it explodes. By substituting this value into the given function rule, we can find its maximum height.
The firework will explode at 1152 feet. What a height!
Consider a parabola with an unknown equation.
We know that the x-coordinate of the vertex defines the axis of symmetry of the parabola.
Vertex | Axis of Symmetry |
---|---|
( 2,5) | x= 2 |
Therefore, the axis of symmetry is the vertical line x= 2. The point ( -1,-3) is 3 units to the left from the axis of symmetry. On the other side of the axis of symmetry, there is a point that lies on the parabola which is also 3 units away.
Note that the x-coordinate of this point is 5. This means that ( 5, -3) is another point on the parabola.
We are given two points that lie on the parabola, (0, 4) and (4, 4). Notice that the y-coordinates of the points are both equal to 4.
Recall that the axis of symmetry is a vertical line that divides the parabola in two mirror images. This means that it should lie midway between the points. Consequently, (0,4) and (4,4) lie on the opposite sides of the axis of symmetry. Let's mark the distance d from each point to the axis in the diagram.
From the diagram, we can see that the distance between the points is equal to 2d. Because the y-coordinates of the points are equal, this distance is the difference between the x-coordinates of the points. 2d = 4 - 0 ⇕ d = 2 Now, the equation of the axis of symmetry can by found by adding d=2 to the x-coordinate of the first point.
Since the vertex of the parabola lies on the axis of symmetry, the obtained equation gives us that the x-coordinate of the vertex is 2. Finally, let's graph an example parabola that satisfies the given conditions.
Let's start by recalling the equation of the axis of symmetry of the graph of a quadratic function. ccc Quadratic Function & & Axis of Symmetry [0.8em] y=ax^2+bx+c & & x=- b/2a In the case of b=0, we can find the equation of the axis of symmetry by substituting 0 for b into the formula x=- b2a. Because the expression 2a is in the denominator of the equation, we need to assume that a is not equal to 0.
When b=0, the axis of symmetry is the vertical line x=0.
Once again, let's recall the equation of the axis of symmetry.
x =- b/2a
Notice that the value of c is not included in the equation. This means that the axis of symmetry depends only on the values of a and b. Therefore, c does not influence the equation of the axis of symmetry.