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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
In a suspension bridge, two large cables hang between the towers and are anchored at the road level midway between the towers. The following quadratic function models the shape of the cables.
In this function, x and y are given in feet.
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.
One of the major attractions in Sydney is the Sydney Harbour Bridge. The beams that span between the pillars result in the shape of a quadratic function. The highest point of the bridge is located 85 meters above the road. The road level is 49 meters above the water. The bridge is 400 meters long.
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.
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.