Wize High School Algebra I Textbook (Common Core) > Quadratic Functions

Transformations of Quadratic Functions (Graphing)

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Vertical Stretches and Reflections
Recall that the vertex form of a quadratic equation is y=a(x−h)2+ky=a\left(x-h\right)^2+ky=a(x−h)2+k. Let's explore how these different numbers a, h, and ka,\ h,\ \text{and}\ ka, h, and k affect the graph.

The graphs of y=ax2\bm{\colorOne{y=ax^2}}y=ax2
Based on the following graphs, try to identify the role aaa plays in graphing a quadratic graph.

Write it Down
How does the aaa value affect the graph of y=ax2\bm{y=ax^2}y=ax2?
If aaa is positive, the graph           opens up          
If aaa is negative, the graph           opens down          
The transformation is                     a vertical reflection along the x-axis                    
If a>1a>1a>1 or a<−1a<-1a<−1, the graph                     is stretched so it's longer                    
If 0<a<10<a<10<a<1 or −1<a<0-1<a<0−1<a<0, the graph                     is compressed (shrunk) so it's shorter                    
The transformation is                     a vertical stretch or compression                    

Practice: Vertical Reflections
Select all of the equations that results in a parabola (U-shape) that opens down.

A) y=x2−3y=x^2-3y=x2−3

B) y=−x2−3y=-x^2-3y=−x2−3

C) y=−2x2y=-2x^2y=−2x2

D) y=2(x−2)2+1y=2(x-2)^2+1y=2(x−2)2+1

E) y=−0.5(x+1)2−5y=-0.5(x+1)^2-5y=−0.5(x+1)2−5

F) y=−34(x−1)2+53y=-\frac{3}{4}(x-1)^2+\frac{5}{3}y=−43​(x−1)2+35​

I don't know

Practice: Vertical Stretch & Compressions
For each of the following equations, identify the type of vertical stretch or compression that is being applied (if there are any)

y=2(x+3)2−5y=2(x+3)^2-5y=2(x+3)2−5

A) Vertical stretch by a factor of 222

B) Vertical compression by a factor of 12\dfrac{1}{2}21​

C) Vertical stretch by a factor of 5

D) Vertical compression by a factor of 15\dfrac{1}{5}51​

E) Neither stretch nor compression

I don't know

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Translations (Shifts)
Recall that the vertex form of a quadratic equation is y=a(x−h)2+ky=a\left(x-h\right)^2+ky=a(x−h)2+k. Let's explore how these different numbers a, h, and ka,\ h,\ \text{and}\ ka, h, and k affect the graph.

The graph of y=x2+k\bm{\colorOne{y=x^2+k}}y=x2+k
Based on the following graphs, try to identify the role kkk plays in graphing a quadratic graph.

Write it Down
How does the kkk value affect the graph of y=x2+k\bm{y=x^2+k}y=x2+k?
If kkk is positive, the graph is                     shifted (translated) up k units                    
If kkk is negative, the graph is                     shifted (translated) down k units                    
The transformation is                     a vertical translation up or down                    

PAGE BREAK
The graph of y=(x−h)2\bm{\colorOne{y=(x-h)^2}}y=(x−h)2
Based on the following graphs, try to identify the role hhh plays in graphing a quadratic graph.


Write it Down
How does the hhh value affect the graph of y=(x−h)2\bm{y=(x-h)^2}y=(x−h)2?
If hhh is positive, the graph is                     shifted (translated) h units to the right                    
We will see a negative number within the brackets like y=(x−3)2y=(x-3)^2y=(x−3)2
If hhh is negative, the graph is                     shifted (translated) h units to the left                    
We will see a positive number within the brackets like y=(x+3)2y=(x+3)^2y=(x+3)2
The transformation is                     a horizontal translation right or left                    

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Example: Translations
Determine the equation of the orange graph above.

The graph is shifted 2 units to the right and 3 units up, so the equation is y=(x+2)2+3y=(x+2)^2+3y=(x+2)2+3.

Practice: Translations
Determine the equation of the graph that results in applying the following transformations to the graph of y=x2y=x^2y=x2.

a) Vertical shift up by 3 units

b) Vertical shift down by 2 units

c) Horizontal shift to the right by 5 units

d) Horizontal shift to the left by 4 units

a) Enter the equation in the form

y=...

b) Enter the equation in the form

y=...

c) Enter the equation in the form

y=...

d) Enter the equation in the form

y=...

I don't know

Practice: Translations
Determine the equation of the following graphs.

a) 

b)

c)

a) Enter the equation in the form

y=...

b) Enter the equation in the form

y=...

c) Enter the equation in the form

y=...

I don't know

Wize High School Algebra I Textbook (Common Core) > Quadratic Functions

Transformations of Quadratic Functions (Graphing)

Popular Courses

Vertical Stretches and Reflections

The graphs of $\bm{\colorOne{y=ax^2}}$

Practice: Vertical Reflections

Practice: Vertical Stretch & Compressions

Translations (Shifts)

The graph of $\bm{\colorOne{y=x^2+k}}$

The graph of $\bm{\colorOne{y=(x-h)^2}}$

Example: Translations

Practice: Translations

Practice: Translations