UNIT 5 • QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations...
Transcript of UNIT 5 • QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations...
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource U5-156
© Walch Education
IntroductionQuadratic functions are used to model various situations. Some situations are literal, such as determining the shape of a parabola, and some situations involve applying the key features of quadratics to real-life situations. For example, an investor might want to predict the behavior of a particular mutual fund over time, or an NFL scout might want to determine the maximum height of a ball kicked by a potential football punter.
In this lesson, we will look specifically at the vertex form of a quadratic, f(x) = a(x – h)2 + k, where the vertex is the point (h, k). The vertex can be read directly from the equation.
Key Concepts
• Standard form, intercept form, and vertex form are equivalent expressions written in different forms.
• Standard form: f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term
• Intercept form: f(x) = a(x – p)(x – q), where p and q are the zeros of the function
• Vertex form: f(x) = a(x – h)2 + k, where the vertex of the parabola is the point (h, k)
• To identify the vertex directly from an equation in vertex form, identify h (the x-coordinate of the vertex) and k (the y-coordinate of the vertex).
• Note that the original equation in vertex form has the quantity x – h, so if the equation has a subtraction sign then the value of h is h.
• This is true because x – (–h) simplifies to x + h.
• However, if the quantity is written as x + h, the value of h is –h. A quadratic function in standard form can be created from vertex form, f(x) = a(x – h)2 + k, where (h, k) is the vertex of the quadratic.
• To do so, distribute and simplify by combining like terms.
Prerequisite Skills
This lesson requires the use of the following skills:
• identifying the standard form of a quadratic function
• finding the vertex of a quadratic function
• solving equations by taking the square root
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU5-157
• For example, f(x) = 3(x – 2)2 + 4 becomes f(x) = 3x2 – 12x + 16.
• A quadratic function in vertex form can be created from standard form, f(x) = ax2 + bx + c.
• To do so, complete the square, or determine the value of c that would make ax2 + bx + c a perfect square trinomial.
• To complete the square, take the coefficient of the linear term, divide by the product of 2 and the coefficient of the quadratic term, and square the quotient.
ax2 + bx + c
a xb
ax
b
a
b
ac2
2 2
2 2+ +
−
+
a xb
aa
b
ac+
−
+2 2
2 2
• Since the quotient of b and 2a is a constant term, we can combine it with the constant c to get
the equation a xb
ak+
+2
2
, where k ab
ac= −
+2
2
.
• For example, f(x) = 2x2 – 12x + 22 becomes f(x) = 2(x – 3)2 + 4.
• When graphing a quadratic using vertex form, if the vertex is the y-intercept, choose two pairs of symmetric points to plot in order to sketch the most accurate graph.
Common Errors/Misconceptions
• forgetting to make sure the coefficient of the quadratic term, x2, is 1 before completing the square
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource U5-158
© Walch Education
Example 1
Given the quadratic function f x x( )= +( ) −1
26 22 , identify the vertex and determine whether it is a
minimum or maximum.
1. Identify the vertex.
The vertex form of a quadratic is f(x) = a(x – h)2 + k, with the vertex (h, k).
f x x( )= +( ) −1
26 22 is in vertex form; therefore, the vertex of this
function is (–6, –2).
2. Determine if the vertex is a minimum or maximum.
If a > 0, the quadratic opens up and thus has a lowest point or a minimum.
If a < 0, the quadratic opens down and thus has a highest point or a maximum.
f x x( )= +( ) −1
26 22 is in vertex form, so a =
1
2.
1
20> , so the vertex of the function is a minimum.
Guided Practice 5.3.3
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU5-159
Example 2
Determine the equation of a quadratic function that has a minimum at (–4, –8) and passes through the point (–2, –5).
1. Substitute the vertex into f(x) = a(x – h)2 + k.
f(x) = a(x – h)2 + k Vertex form
f(x) = a[x – (–4)]2 + (–8) Substitute (–4, –8) for h and k.
f(x) = a(x + 4)2 – 8 Simplify.
2. Substitute the point (–2, –5) into the equation from step 1 and solve for a.
f(x) = a(x + 4)2 – 8 Equation
–5 = a[(–2) + 4]2 – 8 Substitute (–2, –5) for x and f(x).
–5 = a(2)2 – 8 Simplify.
–5 = 4a – 8
3 = 4a
3
4= a
3. Substitute a into the equation from step 1.
f(x) = a(x + 4)2 – 8
f x x( )= +( ) −3
44 82
The equation of the quadratic function with a minimum at (–4, –8)
and passing through the point (–2, –5) is f x x( )= +( ) −3
44 82 .
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource U5-160
© Walch Education
Example 3
Convert the function g(x) = –7x2 + 14x to vertex form.
1. Complete the square to rewrite the function in vertex form.
g(x) = –7x2 + 14x Original function
g(x) = –7(x2 – 2x) Factor out –7.
g(x) = –7(x2 – 2x + 1 – 1) Complete the square.
g(x) = –7(x2 – 2x + 1) + 7 Rewrite the equation as a perfect square trinomial.
g(x) = –7(x – 1)2 + 7 Rewrite the equation as a binomial squared.
2. Summarize your result.
The function g(x) = –7x2 + 14x written in vertex form is g(x) = –7(x – 1)2 + 7.
Example 4
Sketch a graph of the quadratic function y = (x + 3)2 – 8. Label the vertex, the axis of symmetry, the y-intercept, and one pair of symmetric points.
1. Identify the vertex and the equation of the axis of symmetry.
Given the vertex form of a quadratic function, f(x) = a(x – h)2 + k, the vertex is the point (h, k).
The vertex of the quadratic y = (x + 3)2 – 8 is (–3, –8).
The axis of symmetry extends through the vertex.
The equation of the axis of symmetry is x = –3.
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU5-161
2. Find the y-intercept.
The parabola crosses the y-axis when x = 0.
Substitute 0 for x to find y.
y = (x + 3)2 – 8 Original equation
y = (0 + 3)2 – 8 Substitute 0 for x.
y = 32 – 8 Simplify.
y = 1
The y-intercept is the point (0, 1).
3. Find an extra point to the left or right of the axis of symmetry.
Choose an x-value and substitute it into the equation to find the corresponding y-value.
Typically, choosing x = 1 or x = –1 is simplest arithmetically, if these numbers aren’t already a part of the vertex or axis of symmetry.
In this case, let’s use x = 1.
y = (x + 3)2 – 8 Original equation
y = (1 + 3)2 – 8 Substitute 1 for x.
y = 42 – 8 Simplify.
y = 8
The parabola passes through the point (1, 8).
x = 1 is 4 units to the right of the axis of symmetry, x = –3.
4 units to the left of the axis of symmetry and horizontal to (1, 8) is the symmetric point (–7, 8).
UNIT 5 • QUADRATIC FUNCTIONSLesson 3: Creating Quadratic Equations in Two or More Variables
Instruction
CCGPS Analytic Geometry Teacher Resource U5-162
© Walch Education
4. Plot the points you found in steps 2 and 3 and their symmetric points over the axis of symmetry.
x
y
–8 –6 –4 –2 0 2 4 6 8
–10
–8
–6
–4
–2
0
2
4
6
10
8Axis of symmetryx = –3(–7, 8) (1, 8)
(–6, 1) (0, 1)
V (–3, –8)