Math Problem Statement

Find the x and y intercepts of the given function: y = 2∛(x - 8) + 2

Solution

The given function is:

y=2x83+2y = 2 \sqrt[3]{x - 8} + 2

To find the x-intercept and y-intercept, we follow these steps:

1. Finding the y-intercept:

The y-intercept occurs when x=0x = 0. Substitute x=0x = 0 into the equation and solve for yy:

[ y = 2 \sqrt[3]{0 - 8} + 2 = 2 \sqrt[3]{-8} + 2 = 2(-2) + 2 = -4 + 2 = -2 ]

Thus, the y-intercept is at (0,2)(0, -2).

2. Finding the x-intercept:

The x-intercept occurs when y=0y = 0. Set y=0y = 0 and solve for xx:

0=2x83+20 = 2 \sqrt[3]{x - 8} + 2

First, subtract 2 from both sides:

2=2x83-2 = 2 \sqrt[3]{x - 8}

Now, divide by 2:

1=x83-1 = \sqrt[3]{x - 8}

Next, cube both sides to eliminate the cube root:

(1)3=x8    1=x8(-1)^3 = x - 8 \implies -1 = x - 8

Solve for xx:

x=7x = 7

Thus, the x-intercept is at (7,0)(7, 0).

Summary:

  • x-intercept: (7,0)(7, 0)
  • y-intercept: (0,2)(0, -2)

Would you like more details or have any questions about this solution?

Here are 5 related questions you could explore:

  1. How would the intercepts change if the function had a different constant term?
  2. What is the domain of the function y=2x83+2y = 2 \sqrt[3]{x - 8} + 2?
  3. How do you find the asymptotes for functions with cube roots?
  4. Can this function have multiple x-intercepts?
  5. How would the graph of the function look, and how would you plot it manually?

Tip: When solving for intercepts, always remember that the y-intercept involves setting x=0x = 0, while the x-intercept involves setting y=0y = 0.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cube Roots
Intercepts

Formulas

y-intercept occurs at x = 0
x-intercept occurs at y = 0
Solving cube root equations: ∛a

Theorems

Intercept theorem: y-intercept and x-intercept definitions

Suitable Grade Level

Grades 10-12