Cube Root Functions - More Functions - High School Algebra II Unlocked (2016)

High School Algebra II Unlocked (2016)

Chapter 6. More Functions

GOALS

By the end of this chapter, you will be able to


•Graph cube root functions

•Graph piecewise-defined functions, including step functions (such as floor and ceiling functions) and absolute value functions

•Solve single-variable absolute value equations, both algebraically and by graphing systems of equations

•Graph exponential functions

•Write and use cube root, piecewise-defined, and exponential functions to represent real-life situations and solve problems

Lesson 6.1. Cube Root Functions

REVIEW

For any function f(x), the graph of f(x + k) is a horizontal translation of f(x) by k units (to the left if k is positive and to the right if k is negative), and f(x) + k is a vertical translation of f(x) by k units (upward if k is positive, downward if k is negative), where k is a constant.

For any function f(x), the graph of −f(x) is a reflection of f(x) across the x-axis, and the graph of f(−x) is a reflection of f(x) across the y-axis.

When a graph is symmetrical with respect to a point, each point in the graph is the same distance from the center point as a point directly opposite it (on a line passing through the center point). If the image is rotated 180° around the center point, it exactly matches the original image.

Cube root functions are, like square root functions, another type of radical function. They are the inverse of cubic functions (sometimes requiring a domain restriction).

The cubic function y = x3 − 2 is shown on the coordinate grid below.

This is similar to what
we saw in Example 16
in Lesson 3.6, where
we found a square
root function as the
inverse of a quadratic
function (with a
domain restriction).

To find the inverse relationship, switch the x and y variables, then solve for the new y.

x = y3 − 2

Switch the x and y in y = x3 − 2.

x + 2 = y3

Add 2 to both sides.

= y

Take the cube root of both sides.

See Lesson 5.1 for
a review of inverse
functions, including
how to algebraically
find the inverse
relation for a given
function, and how
the graphs of inverse
functions are related.

The inverse function of y = x3 − 2 is the cube root function y = . Inverse functions are reflections of one another across the diagonal line y = x, so the graph of y = is the reflection of y = x3 − 2 across the line y = x.

The parent cube root function, f(x) = , is the inverse of the parent cubic function, f(x) = x3. The graph of f(x) = is shown below.

The sign (positive or
negative) is preserved
when you cube a
number, so the cube
root will automatically
give the correct sign.
This means that we
do not need to use
the ± symbol that is
necessary when taking
the square root of both
sides of an equation
of the form y2 = a.

Comparing the previous graph to this one, you can see that y = is the function f(x) = translated 2 units to the left.

Graph the function g(x) = + 1. What are its x- and y-intercepts?

The function g(x) = + 1 is the function f(x) = translated 8 units to the right and 1 unit up.

We could also create
a table of values to
find points on the
graph. The value of
g(0) = −1, the value
of g(7) = 0, the value
of g(8) = 1, and the
value of g(9) = 2.

If we have graphed accurately or used graphing technology, we can visually identify the x- and y-intercepts. The graph has an x-intercept of 7 and a y-intercept of −1. Let’s use the function equation to confirm that these intercepts are correct.

The x-intercept occurs when y = 0, or g(x) = 0.

0 = + 1

Substitute 0 for g(x) in the function equation.

−1 =

Subtract 1 from both sides, to isolate the cube root.

(−1)3 = x − 8

Cube both sides.

−1 = x − 8

Evaluate (−1)3.

7 = x

Add 8 to both sides.

The x-intercept is indeed 7.

The y-intercept occurs when x = 0.

g(x) = + 1

Substitute 0 for x in the function equation.

g(x) = + 1

Simplify within the radical.

g(x) = −2 + 1

Evaluate .

g(x) = −1

Add −2 and 1.

The y-intercept is −1, as we expected.

The function g(x), like the other two cube root functions we have seen so far, is always increasing.

A cube root function of the form f(x) = a + c is either always increasing or always decreasing. It has a domain of all real numbers and a range of all real numbers. It has exactly one x-intercept and exactly one y-intercept, although in some cases those occur at the same point (the origin).

The parent cube
root function,
f(x) = , is an
example of a cube
root function whose
x-intercept is the same
as its y-intercept:
the point (0, 0).

Graph the functions s(x) = 4 and t(x) = , and compare them to each other and to y = , shown previously.

Let’s create a table of values for each of these functions. Choose x-values that will be easy to work with when evaluating each of the cube root expressions.

The graphs of s(x) = 4 (passing through (−8, −8), (−1, −4), (0, 0), (1, 4), and (8, 8)) and t(x) = (passing through (−2, −2), (−1/4, −1), (0, 0), (1/4, 1), and (2, 2)) are shown below.

Although we are not
showing the point
(16, 4) on the graph of
t(x), it gives us a sense
of the rate at which the
function is increasing.
We could also estimate
the value of t(x) at
x = 8, to approximate
the extension of
the graph to here:

≈ 3.2. The function
t(x) passes through a
point near (8, 3.2).

Both of these functions increase more quickly than y = , so their graphs are more vertically spread out. The graph of s(x) = 4 increases more quickly than that of t(x) = . This makes sense, considering that is the same as , or approximately 1.59. The value of 4 is greater than 1.59 for all x-values greater than 0.

The coefficient a of a cube root function f(x) = a affects the scale of the graph. For |a| > 1, the function increases (or decreases) more quickly than y = , and for 0 < |a| < 1, the function increases (or decreases) more slowly. This is also the case for translated versions of this cube root function. When a < 0 (when a is negative), the function is reflected across the x-axis.

Note that any cube root function of the form f(x) = a is an odd function, meaning that f(−x) = −f(x) (or a = −a). In other words, a reflection of f(x) = a across the y-axis produces the same graph as a reflection across the x-axis. However, when the function is of the form f(x) = a + c and a < 0, the translated function is reflected across the line y = c or across the line x = b. You can also accomplish this by reflecting the graph across the appropriate axis before translating as indicated, with the same result.

The graph of f(x) =
a produces the
same image when
reflected across
the y-axis as when
reflected across the
x-axis because it is
symmetrical with
respect to the origin
(as is a cubic function
of the form f(x) = ax3,
as well as any other
odd function). If the
image is rotated 180°,
it exactly matches
the original image.

Graph the function f(x) = .

This function should be the graph of y = translated 5 units to the left, reflected across the x-axis (or reflected across the line x = −5), and stretched vertically by a factor of 2. Let’s create a table of values to find points on the graph.

The function f(x) = passes through the points (−13, 4), (−6, 2), (−5, 0), (−4, −2), and (3, −4), as shown below.

The graph matches our prediction: It is the graph of y = translated 5 units to the left, reflected across the x-axis (or across the line x = −5), and stretched vertically by a factor of 2.

In Lesson 4.4, when
looking at tangent
curves, we learned that
the point of inflection
is the point where
a curve changes its
direction, from concave
to convex or from
convex to concave.
You can also view the
point of inflection
in these cube root
function graphs as
the point where the
graph changes from
concave upward to
concave downward,
or vice versa.

Notice that this graph is symmetrical with respect to the point (−5, 0), which is the point of inflection for this cube root function graph.

The point of inflection of a cube root function graph of the form f(x) = a + c is the point (b, c). The graph is symmetrical with respect to this point.

This is why a reflection across the line y = c produces the same result as a reflection across the line x = b.

In Example 3, b = −5 and c = 0 in the form f(x) = a + c, so a reflection across the line x = −5 has the same effect as a reflection across the line y = 0, also known as the x-axis.

If p(x) = x3 + 3x2 − 4x − 12, graph its inverse relation. Is this inverse relation a cube root function?

For p(x), it is not easy to solve for its inverse. If we set x = y3 + 3y2 − 4y − 12, we cannot simply solve for the new y. Instead, let’s graph p(x) = x3 + 3x2 − 4x − 12 and then reflect that graph across the line y = x.

If we factor x2 out of the first two terms and −4 out of the last two terms, each pair of remaining terms is (x + 3).

x2(x + 3) − 4(x + 3)

Factor each pair of terms in x3 + 3x2 − 4x − 12.

(x2 − 4)(x + 3)

Use the distributive property to factor out (x + 3).

(x − 2)(x + 2)(x + 3)

Factor x2 − 4, a difference of squares.

So, p(x) = x3 + 3x2 − 4x − 12 = (x − 2)(x + 2)(x + 3), so it has x-intercepts of 2, −2, and −3. It is a cubic with a positive leading coefficient, so its left arm points down and its right arm points up. When x = 0, p(x) = −12, so its y-intercept is −12. We can also use the equation to solve for a few more points on the graph.

p(−4) = (−4)3 + 3(−4)2 − 4(−4) − 12 = −64 + 48 + 16 − 12 = −12
p(−1) = (−1)3 + 3(−1)2 − 4(−1) − 12 = −1 + 3 + 4 − 12 = −6
p(1) = 13 + 3(12) − 4(1) − 12 = 1 + 3 − 4 − 12 = −12
p(3) = 33 + 3(32) − 4(3) − 12 = 27 + 27 − 12 − 12 = 30

Alternatively, if
we did not see the
pattern in the original
cubic, we could have
used the Rational
Root Theorem, the
Remainder Theorem,
and long division of
polynomials to fully
factor the cubic, as
we did in Example
18 in Lesson 1.5.

The graph of p(x) = x3 + 3x2 − 4x − 12 is shown below.

When we reflect this graph across the line y = x, each point (a, b) will be mapped to a point (b, a). So, the inverse relation of p(x) will pass through the points (−12, −4), (0, −3), (0, −2), (−6, −1), (−12, 0), (−12, 1), and (0, 2), as shown below.

This graph fails the vertical line test, so the relation x = y3 + 3y2 − 4y − 12 is not a function.

However, if we set constraints on the domain of the function p(x), then its inverse will be a function. The portion of p(x) that is between x = −2 and x = 0 is always decreasing, meaning that only one x-value is associated with each y-value here, so the section of the graph of the inverse relation between y = −2 and y = 0 is a function, with only one y-value associated with each x-value. So, if p(x) = x3 + 3x2 − 4x − 12 for −2 ≤ x ≤ 0, then p−1(x) is a function, as shown below.

Alternatively, we could
have defined p(x) to
have a domain of only
x ≥ 1, where the graph
is always increasing,
and this would create
an inverse that is also
a function. Or, we
could have chosen the
section where x ≤ −3.