# Lesson 7

Expressing Transformations of Functions Algebraically

• Let’s express transformed functions algebraically.

### 7.1: Describing Translations

Let $$g(x)=\sqrt{x}$$. Complete the table. Be prepared to explain your reasoning.

words (the graph of $$y=g(x)$$ is...) function notation expression
translated left 5 units $$g(x+5)$$
translated left 5 units and down 3 units   $$\sqrt{x+5}-3$$
$$g(\text-x)$$ $$\sqrt{\text-x}$$
translated left 5 units, then down 3 units,
then reflected across the $$y$$-axis

### 7.2: Translating Vertex Form

Let $$f$$ be the function given by $$f(x) = x^2$$.

1. Write an equation for the function $$g$$ whose graph is the graph of $$f$$ translated 3 units left and up 5 units.
2. What is the vertex of the graph of $$g$$? Explain how you know.
3. Write an equation for a quadratic function $$h$$ whose graph has a vertex at $$(1.5, 2.6)$$.
4. Write an equation for a quadratic function $$k$$ whose graph opens downward and has a vertex at $$(3.2, \text-4.7)$$.

### 7.3: An Even Better Fit

In an earlier lesson, we looked at the temperature $$T$$, in degrees Fahrenheit, of a bottle of soda water left outside for $$h$$ hours. Let’s model this data with a function. This time, we will start with the function $$f(h) = 33(0.6)^{h}$$. This graph has a shape that fits the data well.

1. Describe a translation of this graph that fits the data.
2. Write an equation defining a function $$g$$ that models the data.

Han tried the following steps to model the soda water temperature. First he shifts the given graph left by one hour, then he applies a vertical shift.

1. What vertical shift does Han need to apply to model the 45 degree Fahrenheit temperature in the refrigerator?
2. How does Han’s model compare to yours?

### Summary

You can use the equation of a function to write an equation for its transformation. For example, let $$f(x) = x^2$$. Take the graph of $$f$$, reflect it across the $$x$$-axis, translate it up 10 units, and translate it left 3 units. What is an equation for this new function? The new function $$g$$ is related to $$f$$ by $$g(x) =\text-f(x+3)+10$$, since

Which means $$g(x) =\text-(x+3)^2 + 10$$.

Sometimes you can recognize from the expression for a function that it is the transformation of a simpler function. For example, consider:

$$\displaystyle H(t) = 10 - (1.2)^{t+5}$$

One way to obtain the expression for $$H$$ from $$1.2^t$$ is:

• adding 5 to the input to get $$(1.2)^{t+5}$$
• multiplying the output by -1 to get $$\text-(1.2)^{t+5}$$
• adding 10 to the output to get $$10 - (1.2)^{t+5}$$

So the graph of $$H$$ is obtained from the graph of $$f(t) = 1.2^t$$ by translating left 5 units, reflecting across the $$x$$-axis, and translating up 10 units. Consider the point $$(0,1)$$ on the graph of $$f$$. After translating, reflecting, and translating again, it becomes the point $$(\text-5,9)$$ on the graph of $$H$$.

### Glossary Entries

• even function

A function $$f$$ that satisfies the condition $$f(x) = f(\text-x)$$ for all inputs $$x$$. You can tell an even function from its graph: Its graph is symmetric about the $$y$$-axis.

• odd function

A function $$f$$ that satisfies $$f(x) = \text-f(\text-x)$$ for all inputs $$x$$. You can tell an odd function from its graph: Its graph is taken to itself when you reflect it across both the $$x$$- and $$y$$-axes. This can also be seen as a 180$$^\circ$$ rotation about the origin.