## Key Concepts

- Reflections of functions across the “x-axis”.
- Vertical stretches and compressions of graphs of functions.
- Horizontal stretches and compressions of graphs of functions.

## Quadratic functions & its characteristics

The **quadratic parent function** is 𝒇𝒙=𝒙𝟐

It is the simplest function in quadratic function family. The graph of the function is a curve called a **parabola**.

The **vertex** is the lowest/highest point on the graph of a quadratic function.

The vertex form of a quadratic function is 𝒇𝒙=𝒂(𝒙−𝒉)𝟐+𝒌 **.**

The graph of f is the graph of 𝒈𝒙=𝒂𝒙𝟐 translated **horizontally** 𝒉** units and vertically** 𝒌** units**.

The vertex is located at 𝒉, 𝒌.

The axis of symmetry is 𝒙=𝒉.

The vertex form of a quadratic function is 𝒇𝒙=𝒂(𝒙−𝒉)𝟐+𝒌.

The graph of f is the graph of

g(x)=ax2𝒈𝒙=𝒂𝒙𝟐 translated **horizontally** 𝒉** units and vertically** 𝒌** units**.

The vertex is located at 𝒉, 𝒌.

The axis of symmetry is𝒙=𝒉**.**

### Translations of functions

#### Vertical translation

For a given

g(x)=f(x)+k, the graph of the function g is the function f translates k units vertically.

k>0: shifts |k|units up

k<0: shifts |k| units down

#### Horizontal translation

For a given

g(x)=f(x−h), the graph of the function g is the function f translates h units horizontally.

h>0: shifts |h|units right

h<0: shifts |h| units left

#### Combined translation

For a given

g(x)=f(x−h)+k, the graph of the function g is the graph of function f translated h units horizontally, then translated k units vertically.

**Example: **

How does the function

f(x)=x2 transform to the function g(x)=(x−3)2+2.** **Also find the vertex of the function g(x).

**Solution: **

Graph of the function

f(x)=x2 and g(x)=(x−3)2+2.

The function g(x) is of the form (x−h)2+k(x−h)2+k has a translation of h units horizontally and k units vertically.

The function f(x)=x2 transforms to the function g(x)=(x−3)2+2:

3 units right horizontally and 2 units up vertically.

The vertex of the function g(x) is (3, 2).

### Reflections across the x-axis

**Example 1: **

Consider the function

f(x)=√fx=x

and

g(x)=√gx=−x

The graph of g(x)=√x is a reflection of f(x)=√x across the x−axis.

**Example 2: **

Consider the function

f(x) = x^{2}

and

g(x) =−x^{2}

The graph of g(x)=−x^{2} is a reflection of f(x)=x^{2} across the x−axis.

In general, g(x)=−1f(x) , the graph of g is a reflection across the x−axis of the graph of f.

So, for any function when the output is multiplied by −1 it reflects across the x−axis.

### Vertical compressions and stretches of graph

#### Vertical stretches of graphs

**Example: **

Consider g(x)=kf(x) for |k|>1k>1 when function f(x)=x^{2}.

**Solution: **

Here is the graph of g is a vertical stretch away from the x−axis of the graph of f.

So, the graph of g(x)=2x^{2} is vertical stretch of f(x)=x^{2} away from the x−axis.

**Example: **

Consider g(x)=kf(x) for |k|>1k>1 when function fx=x^{3}

**Solution:**** **

Here is the graph of g is a vertical stretch away from the x−axis of the graph of f.

So, the graph of g(x)=2x^{3} is vertical stretch of f(x)=x^{3 }away from the x−axis.

#### Vertical compressions of graphs

**Example: **

Consider g(x)=kf(x) for 0<|k|<1 when function f(x)=|x+1|.

**Solution: **

Here is the graph of g is a vertical compression towards the x−axis of the graph of f.

So, the graph of g(x)=1/2 |x+1| is vertical compression of f(x)=|x+1| towards the x−axis.

**Example: **

Consider g(x)=kf(x) for 0<|k|<1 when function f(x)=x^{2}.

**Solution: **** **

Here is the graph of g is a vertical compression towards the x−axis of the graph of f.

So, the graph of g(x)=1/2 x^{2} is vertical compression of f(x)=x^{2} towards the x−axis.

#### Vertical compressions & stretches of graphs

Given a function,

f(x) , a new function g(x)=kf(x), where a is a constant, is vertical stretch or a vertical compression of the function f(x).

- If |k|>1k>1, then the graph will be stretched.
- If 0<|k|<10<k<1, then the graph will be compressed.

### Horizontal compressions and stretches of graphs

#### Horizontal stretches of graphs

**Example: **

Consider g(x)=f(kx) for 0<k<1 when function f(x)=x^{2}

**Solution: **

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f.

So, the graph of gx= (1/2 x)^{2 }is horizontal compression of fx=x^{2} toward the y-axis.

**Example: **

Consider g(x) = f(kx) for [0<k<1] when function f(x) = √x

**Solution: **

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f.

So, the graph of g(x)= 1/4 x is horizontal compression of f(x)=√x toward the y-axis.

#### Horizontal compressions of graphs

**Example: **

Consider g(x)=f(kx) for k>1 when function f(x)=(x−1)^{2}

**Solution: **

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f.

So, the graph of g (x)= (3x-1)^{2} is horizontal compression of f(x)= (x-1)^{2 }toward the y-axis.

**Solution: **

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f.

So, the graph of g(x) = |2x+1| is horizontal compression of f(x) = |x+1| toward the [y-axis].

#### Horizontal compressions & stretches of graphs

Given a function [f(x)], a new function gx = f(kx), where k is a constant, is a horizontal stretch or a horizontal compression of the function f(x).

- If [|k|<1], then the graph will be compressed.
- If [0<|k|<1], then the graph will be stretched.

## Exercise

- Write a function with a graph that is the reflection of the graph of fx=(x-1)
^{2}across the x-axis - For each pair, identify the graph of g is a vertical or horizontal compression or stretch of the graph of f.
- f(x)=|3x-1| , g(x) = 1/4 |3x-1|
- f(x) = √x , g(x) = √1/5x
- f(x)=√x-4 , g(x)=√3x-4
- f(x) = x²+4 , g(x)=2x²+8

- Write a function with a graph that is a vertical stretch of the graph of fx=|x|, away from the x-axis.
- Write a function with a graph that is a horizontal compression of the graph of fx=3x, toward the y-axis.

### Concept Summary

### What we have learned

- Reflections across the x-axis.
- Vertical stretches of graphs.
- Vertical compressions of graphs.
- Horizontal stretches of graphs.
- Horizontal compressions of graphs.

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