Parent Functions and Transformatio

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Parent Graphs

• In the previous lesson you discussed the following functions:• Linear• Quadratic• Cubic, Square root• Cube root• Absolute value• Exponential

• In this lesson we are going to learn how these functions can transform

Parent Graph of a Quadratic Function

Lets look at the graph of this function. First we are going to make an input-output table of values and then we are going to graph the function.

On the next slide we are going to see a transformation of this function.

Vertical Transformation

Now lets look at the graph of this function.

Lets make another input output table.

Compare the y values of this slide to the previous slide. What happened to the y values? What would happen to the y values if the function was ? The answer to these questions are on the next slide.

Vertical Transformation

The y values all increase by 4 units. You can also see that the graph shifted up 4 units. This shows that if I add a number to a parent function then it shifts the entire graph up that many units.

If we subtract 4 from then all the y values will reduce by 4 units, meaning that the graph will shift down 4 units.

Horizontal Transformation

Now lets look at the graph of this function.

We are just going to look at the graph in this example.

What happened from the parent graph to this graph?

This graph shifted to the right 3 units.

What do you think would happen if you graphed ?

Horizontal Transformation

Did you Guess correctly?

What happened from the parent graph to this graph?

This graph shifted to the left 3 units.

Please note that when we put the number inside the parenthesis, it shifted the graph horizontally. When we put the numbers at the end of the entire function, if shifted it vertically.

Vertical & Horizontal Transformation

Compared to the parent function of a quadratic function? How does the following function transform ?

This graph shifted to the lest 2 units and down 6 units.

The +2 moved it left 2

The -6 moved it down 6

Vertical Stretch and Horizontal Compress

Now that we have seen the vertical and horizontal movements of a quadratic function. Lets look at some transformations that can cause a vertical stretch or compression in the x direction.

Reflection Across the X-axis

Compared to the parent function of a quadratic function? How does the following function transform from the parent graph? (Go back and look at the 3rd slide at the parent graph)

You can see that the y values were all made into a negative. This made the graph reflect across the x-axis.

Do you know what would happened if we graphed ?

Reflection Across the y-axis

Lets see what happed if I put a negative on the x coordinate?

This reflection is hard to see as a difference from the parent graph BUT this function reflected across the y –axis (this is easier to see by using another parent graph).

In general, if you put a negative on the outside of the x coordinate it will flip it across the x-axis. If you put a negative on the x coordinate, then that reflects it across the y-axis.

Vertical Stretch

Compared to the parent function of a quadratic function? How does the following function transform from the parent graph? (Go back and look at the 3rd slide at the parent graph)

You can see that the y values were all tripled. This made the graph become more narrow compared to the parent graph. This is going to be called a vertical stretch by a factor of 3. What is going to happen if I use a 1/3 instead of a 3?

Vertical Shrink

Lets see what happed if I use a 1/3?

You can see that the y values were all reduced. Meaning this made the graph become less narrow compared to the parent graph. This is going to be called a vertical shrink.

Other Parent Graphs

• In the previous slides, we discuss the different transformations with a quadratic function.

• The transformations that we discussed will transfer to EVERY parent graph.

• Look at the next slide to see a different parent graph being used.

Other Parent Graphs

• For example if you have the parent function of a quadratic, how would the following graph compare to it.

• If we use the parent graph of an exponential function how would the following graph compare to it.

Vertical Stretch by a factor of 4

The negative reflects it across the x-axis.

Shifts right 5

Shifts up 1

Vertical shrink by a factor of 1/3

Shifts left 8

Shifts down 1

General Transformations

• Vertical Shifts

• Horizontal Shifts

• Reflections

General Transformations Continued

• Stretches and Shrinks

The notes didn’t go over this, however the only thing that is different is that if the “c” value is inside the parenthesis, it’s a horizontal shrink or stretch.