CHAPTER 5 5-7 Solving radicals

Objectives Graph radical functions and inequalities. Transform radical functions by changing parameters.

Radical Functions Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs below show the inverses of the quadratic parent function and cubic parent function.

Radical Functions Notice that the inverses of f(x) = x2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x2 to x ≥ 0, its inverse is the function .

Radical Functions A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving . The square-root parent function is . The cube-root parent function is

Example#1 Graph each function and identify its domain and range. Make a table of values. Plot enough ordered pairs to see the shape of the curve. Because the square root of a negative number is imaginary, choose only nonnegative values for x – 3.

Solution x (x, f(x)) 3 (3, 0) 4 (4, 1) 7 (7, 2) 12 (12, 3)

solution The domain is {x|x ≥3}, and the range is {y|y ≥0}.

Example#2 Graph each function and identify its domain and range. Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

solution x (x, f(x)) –6 (–6, –4) 1 (1,–2) 2 (2, 0) 3 (3, 2) 10 (10, 4)

Solution The domain is the set of all real numbers. The range is also the set of all real numbers

Example#3 Graph each function, and identify its domain and range. x (x, f(x)) –1 (–1, 0) 3 (3, 2) 8 (8, 3) 15 (15, 4)

Solution The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

Graphs of radical functions The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

transformations

Example Using the graph of f(x) =√ x as a guide, describe the transformation and graph the function. g(x) = √x + 5

solution Translate f 5 units up. •

Example Using the graph of f(x) =√ x as a guide, describe the transformation and graph the function. g(x) = √x + 1

Solution Translate f 1 unit up. •

Example Using the graph of f(x) =√ x as a guide, describe the transformation and graph the function. g is f vertically compressed by a factor of 1/2 .

Transformations summarize Transformations of square-root functions are summarized below.

Example of multiple transformations Using the graph of f(x) =√ x as a guide, describe the transformation and graph the function

solution Reflect f across the x-axis, and translate it 4 units to the right. •

Example Using the graph of f(x) =√ x as a guide, describe the transformation and graph the function.

Solution g is f reflected across the y-axis and translated 3 units up. ●

Example g(x) = –3√x – 1 ● g is f vertically stretched by a factor of 3, reflected across the x-axis, and translated 1 unit down.

Writing radical functions Use the description to write the square-root function g. The parent function 𝒇 𝒙 = 𝒙 is reflected across the x-axis, compressed vertically by a factor of 1/5, and translated down 5 units

Solution Step 1 Identify how each transformation affects the function. Reflection across the x-axis: a is negative Vertical compression by a factor of 1/5 Translation 5 units down: k = –5 1 5 a = –

solution Step 2 Write the transformed function. 1 5 g(x) = - x + (- 5) ö ÷ ø æ ç è Substitute –1/5 for a and –5 for k. Simplify.

Example Use the description to write the square-root function g. The parent function 𝑓 𝑥 =√𝑥 is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.

solution Step 1 Identify how each transformation affects the function. Reflection across the x-axis: a is negative Vertical stretch by a factor of 2 Translation 5 units down: k = 1

solution Step 2 Write the transformed function. Substitute –2 for a and 1 for k.

Business Application A framing store uses the function to determine the cost c in dollars of glass for a picture with an area a in square inches. The store charges an addition $6.00 in labor to install the glass. Write the function d for the total cost of a piece of glass, including installation, and use it to estimate the total cost of glass for a picture with an area of 192 in2.

solution Step 1 To increase c by 6.00, add 6 to c. Step 2 Find a value of d for a picture with an area of 192 in2. The cost for the glass of a picture with an area of 192 in2 is about $13.13 including installation.

Graphing radical inequalities In addition to graphing radical functions, you can also graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

Example Graph the inequality

solution Because the value of x cannot be negative, do not shade left of the y-axis.

Example Graph

Solution Because the value of x cannot be less than –4, do not shade left of –4.

Student Guided Practice Do even numbers from 2-20 in your book page 372

Homework DO even numbers from 30-43 in your book page 372

closure Today we learned about radical functions and how to graph and make transformations Next class we are going to solve radical equations