4.2: Graphs of Quadratic Functions in Vertex or Intercept Form

4.2: Graphs of Quadratic Functions in
Vertex or Intercept Form
Objectives:
1. To graph quadratic
functions in vertex
and intercept form
2. To rewrite a
quadratic equation in
standard, vertex, or
intercept form
•
•
•
•
•
Assignment:
P. 249-251: 1, 2, 3-42
M3, 52, 53, 57, 58
P. 256: 45, 48, 51
P. 263: 42, 45, 48
P. 289: 42, 45, 48
Approximating Area
Worksheet
Objective 1a
You will be able to graph a
quadratic function in vertex
form
𝑦 = −0.08 𝑥 − 11.3
2
+ 3.3
Warm-Up
Graph y = 2|x – 3| + 4
using SRT
transformations. How
does this graph
compare with y = |x|?
• S: 2 times y
• R: None
• T: R 3 units, U 4 units
Forms of Quadratic Equations
So far, we’ve been
limiting our discussion
of quadratic functions
to standard form:
y = ax2 + bx + c
We’ll use GSP to
investigate two other
useful forms of
quadratic functions.
Vertex Form
Quadratic Function
in Vertex Form:
y = a(x – h)2 + k
The graph of
y = a(x – h)2 + k is
the graph of y = ax2
translated h units to
the right and k units
up.
Vertex Form
Quadratic Function
in Vertex Form:
y = a(x – h)2 + k
• Vertex: (h, k)
• Axis of Symmetry:
x=h
Exercise 1
What is the y-intercept of the graph of
y = a(x – h)2 + k?
Exercise 2
What are the x-intercepts of the graph of
y = a(x – h)2 + k?
Graphing Quadratic Functions, II
To graph y = a(x – h)2 + k use SRT
transformations.
• Scaling: a times the y-coordinate
• Reflecting: over the x-axis if a < 0
• Translating: Left/Right h units and
Up/Down k units
– Remember that the up/down movement is in
the direction of the sign of k, but for left/right,
it’s the opposite direction as the sign of h.
Exercise 3
Graph
1
2
y   x  3  5
2
Exercise 4
Graph
y    x  2  3
2
Objective 1b
You will be able to graph a
quadratic function in intercept form
𝑦 = −0.42 𝑥 − 16.9 𝑥 − 26.9
Intercept Form
Quadratic Function
in Intercept Form:
y = a(x – p)(x – q)
• x-intercepts: p and q
• Vertex: Average of p
and q
• Axis of Symmetry:
pq
x
2
Exercise 5
What is the y-intercept of the graph of
y = a(x – p)(x – q)?
Graphing Quadratic Functions, III
To graph y = a(x – p)(x – q):
1. Use the a-value to determine  or 
2. Plot the x-intercepts p and q
3. Find and plot the vertex (x = average of p
and q)
4. Graph the axis of symmetry
5. Find and plot a couple of other function
values and their reflections
Exercise 6
Graph
y   x  x  4
Exercise 7
Graph
y  2  x  1 x  4 
𝑦 = −0.05𝑥 2 + 1.2𝑥 − 1.9
𝑦 = −0.05 𝑥 − 12.4
2
+ 5.7
𝑦 = −0.05 𝑥 − 1.6 𝑥 − 23.1
You will be able to rewrite your
quadratic function in standard,
vertex, or intercept form
Objective 2
Exercise 8a
Write each function in standard form.
1. y  7  x  6  x  1
2. y  2  x  3  9
2
Protip #1: Standard Form
To write your quadratic equation in standard
form is really easy. Just FOIL, distribute,
or expand as necessary, and then
combine like terms. Finally, make it look
like this:
y = ax2 + bx + c
Exercise 8b
Write each function in standard form.
1. y  2  x  5 x  4 
2. y  3  x  5  1
2
Exercise 9a
Write each function in intercept form.
1. y  x  5x  14
2
2. y  8x  38x  10
2
Protip #2: Intercept Form
To write your equation in intercept form, you
have to do some factoring:
1. Factor out GCF.
y  6 x 2  x  15
That includes a −1
y  6 x 2  x  15
if the leading term
is negative
y  2 x  33x  5
2. Factor the
trinomial that is left
over
Protip #2: Intercept Form
To write your equation in intercept form, you
have to do some factoring:
3. Each factor has to
look like (x ± p).
This means you
have to divide out
any coefficient of x
from each factor.
Whatever you
divide out becomes
part of the a-value.
y  6 x 2  x  15
y  6 x 2  x  15
y  2 x  33x  5
Divide out a 2
Divide out a 3
Protip #2: Intercept Form
To write your equation in intercept form, you
have to do some factoring:
3. Each factor has to
look like (x ± p).
This means you
have to divide out
any coefficient of x
from each factor.
Whatever you
divide out becomes
part of the a-value.
y  6 x 2  x  15
y  6 x 2  x  15
y  2 x  33x  5
3 
5

y  6 x   x  
2 
3

Exercise 9b
Write each function in intercept form.
1. y  x  2 x  48
2
2
y


12
x
 72 x  105
2.
Exercise 10a
Write each function in vertex form.
1. y  x  8x  17
2
2. y  2 x  24 x  25
2
Protip #3: Vertex Form
To put a quadratic equation in vertex form, you
have to complete the square on only one side of
the equation.
y  4 x 2  24 x  49
1. Separate the
2
y

4
x
 24 x
 49
constant term from
the variable terms
y  4x 2  6 x  ____   ____  49
2. Factor out the
leading coefficient
from the variable
terms
Protip #3: Vertex Form
To put a quadratic equation in vertex form, you
have to complete the square on only one side of
the equation.
y  4 x 2  24 x  49
3. Complete the square
with the variable
terms
4. Whatever you add
on the inside of the
parenthesis, subtract
on the outside of the
parenthesis.
y  4 x 2  24 x
 49
36  49
y  4x 2  6 x  ____
9   ____
Protip #3: Vertex Form
5. Remember that what
you add on the inside
of the parenthesis is
actually multiplied by
the leading coefficient
that you factored out in
Step 2. Make sure to
multiply by this
number before
subtracting on the
outside of the
parenthesis
y  4 x 2  24 x  49
y  4 x 2  24 x
 49
36  49
y  4x 2  6 x  ____
9   ____
y  4  x  3  13
2
Exercise 10b
Write each function in vertex form.
1. y   x  6 x  3
2
2
y

3
x
 9x  4
2.
4.2: Graphs of Quadratic Functions in
Vertex or Intercept Form
Objectives:
1. To graph quadratic
functions in vertex
and intercept form
2. To rewrite a
quadratic equation in
standard, vertex, or
intercept form
Assignment
• P. 249-251: 1, 2, 3-42
M3, 52, 53, 57, 58
• P. 256: 45, 48, 51
• P. 263: 42, 45, 48
• P. 289: 42, 45, 48
• Approximating Area
Worksheet
“Powabowa? That’s a hard word to say.”