Lesson 7

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

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 \((3.2, \text-4.7)\).

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.
3. What does your function tell you about the temperature outside?

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 \(H\).