Math Problem Statement

Complete the table of values for y = x^3 + x - 2 and draw the graph for values of x from -2 to 2.

Solution

The problem asks you to complete the table and graph the function y=x3+x2y = x^3 + x - 2 for xx values from 2-2 to 22.

Step 1: Calculate yy values for xx values from 2-2 to 22.

The equation is: y=x3+x2y = x^3 + x - 2

For x=2x = -2:

y=(2)3+(2)2=822=12y = (-2)^3 + (-2) - 2 = -8 - 2 - 2 = -12

For x=1x = -1:

y=(1)3+(1)2=112=4y = (-1)^3 + (-1) - 2 = -1 - 1 - 2 = -4

For x=0x = 0:

y=(0)3+02=2y = (0)^3 + 0 - 2 = -2

For x=1x = 1:

y=(1)3+12=1+12=0y = (1)^3 + 1 - 2 = 1 + 1 - 2 = 0

For x=2x = 2:

y=(2)3+22=8+22=8y = (2)^3 + 2 - 2 = 8 + 2 - 2 = 8

Step 2: Complete the Table

The completed table is:

\hline x & y \\ \hline -2 & -12 \\ -1 & -4 \\ 0 & -2 \\ 1 & 0 \\ 2 & 8 \\ \hline \end{array}$$ ### Step 3: Plot the Graph On the graph: 1. Plot the points $$(-2, -12)$$, $$(-1, -4)$$, $$(0, -2)$$, $$(1, 0)$$, and $$(2, 8)$$. 2. Connect these points with a smooth curve since $$y = x^3 + x - 2$$ is a cubic function. Would you like further help with plotting or interpreting the graph? ### Related Questions: 1. What is the behavior of cubic functions as $$x \to \infty$$ and $$x \to -\infty$$? 2. How do changes to the constant term in $$y = x^3 + x - 2$$ affect the graph? 3. Can you identify turning points of the function $$y = x^3 + x - 2$$? 4. What is the significance of the $$y$$-intercept in this context? 5. How does the coefficient of $$x^3$$ influence the steepness of the curve? ### Tip: When sketching cubic graphs, remember that they can have up to two turning points, and the end behavior depends on the sign of the leading coefficient.

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Math Problem Analysis

Mathematical Concepts

Cubic Functions
Graphing
Table of Values

Formulas

y = x^3 + x - 2

Theorems

Behavior of polynomial functions

Suitable Grade Level

Grades 9-12