• To explore general patterns and characteristics of cubic functions

• To learn formulas that model the areas of squares and the volumes of cubes

• To explore the graphs of cubic functions and transformations of these graphs

• To write the equation of a cubic function from its graph

•In this lesson you’ll learn about cubic equations, which are often used to model volume. You’ll see that some of the techniques you’ve used with quadratic equations can be applied to cubic equations, too.

• The volume of the cube at right is 64 cubic centimeters cm3, so you can fill the cube using 64 smaller cubes, each measuring 1 cm by 1 cm by 1 cm.

• The edges of a cube have equal length, so you can write its volume formula as volume=(edge length)3.

• The cubing function, f(x)=x3, models volume. Each edge length, or input, gives exactly one volume, or output.

• The edge length of a cube with a volume of 64 is 4. So you can write 43=64.

• You call 4 the cube root of 64 and the number 64 a perfect cube because its cube root is an integer.

• Then you can express the equation as

• You can evaluate cubes and cube roots with your calculator.

34 64

Graphs of cubic functions have different and interesting shapes. In the window -5≤x≤5 and -4≤y≤4, the parent function y=x3 looks like the graph shown.

Example A•Write an equation for each graph.

Rooting for Factors• In this investigation you’ll discover a relationship

between the factored form of a cubic equation and its graph.

• List the x-intercepts for each of these graphs.

•Each equation below matches exactly one graph in the previous step. Use graphs and tables to find the matches.

•Describe how the x-intercepts you found in Step 1 relate to the factored forms of the equations in Step 2.

Now you’ll write an equation from a graph.

• Use what you discovered in Steps 1–3 to write an equation with the same x-intercepts as the graph shown. Graph your equation; then adjust your equation to match the graph.

Example B• Find an equation for the graph

shown.• There are three x-intercepts on the

graph. They are x=0, 1, and 2. • Each intercept helps you find a

factor in the equation. These factors are x, x-1, and x-2.

• Graph the equation y=x(x-1)(x-2) on your calculator.

• The shape is correct, but you need to reflect it across the x-axis. You also need to vertically stretch the graph.

• Check the y-value of your graph at x=1. The y-value is 2. You need it to be 4, so multiply by 2. The correct equation is y=2x(x-1)(x-2). Check this equation by graphing it on your calculator.

• You can also use what you know about roots to convert cubic equations from general form to factored form. Look at the graph of y=x3-3x+2 at right.

• It has x-intercepts x=2 and x=1. Most cubic equations you’ve explored have had three roots. Where is the third one? Notice that the graph just touches the axis at x=1. It doesn’t actually pass through the axis. The root at x=1 is called a double root, and the factor x-1 is squared.

• Graph the factored form y=(x+2)(x-1)2.

Example C• Find the exact x-

intercepts of y=x3+2x2-7x-2.

• The graph shows that 2 is an x-intercept of the function. This means that (x-2) is a factor. You can approximate the other two roots by tracing, but to find exact algebraic solutions you need to factor.

• You can do this using a rectangle diagram.

3 2 22 7 2 2 4 1x x x x x x

3 2 22 7 2 2 4 1x x x x x x

Use the quadratic formula to find the zeros of 2 4 1x x