Math Problem Statement

answer

Complete the table of values for the graph with the equation y = -4x^3 - 12x^2 + 16x - 10 for the given x-values: -2, -1, 0, 1, and 2.

Solution

The table provided corresponds to the equation y=−4x3−12x2+16x−10y = -4x^3 - 12x^2 + 16x - 10. We are asked to complete the table of values by substituting different xx-values into the equation and calculating yy.

The given xx-values are: −2,−1,0,1,2-2, -1, 0, 1, 2.

Let's calculate each yy-value using the equation.

For x=−2x = -2:

= -4(-8) - 12(4) + 16(-2) - 10 = 32 - 48 - 32 - 10 = -58$$ So, $$y = -58$$. ### For $$x = -1$$: $$y = -4(-1)^3 - 12(-1)^2 + 16(-1) - 10 = -4(-1) - 12(1) + 16(-1) - 10 = 4 - 12 - 16 - 10 = -34$$ So, $$y = -34$$. ### For $$x = 0$$: $$y = -4(0)^3 - 12(0)^2 + 16(0) - 10 = -10$$ So, $$y = -10$$. ### For $$x = 1$$: $$y = -4(1)^3 - 12(1)^2 + 16(1) - 10 = -4(1) - 12(1) + 16(1) - 10 = -4 - 12 + 16 - 10 = -10$$ So, $$y = -10$$. ### For $$x = 2$$: $$y = -4(2)^3 - 12(2)^2 + 16(2) - 10 = -4(8) - 12(4) + 16(2) - 10 = -32 - 48 + 32 - 10 = -58$$ So, $$y = -58$$. ### Final Table: $$\begin{array}{|c|c|} \hline x & y \\ \hline -2 & -58 \\ -1 & -34 \\ 0 & -10 \\ 1 & -10 \\ 2 & -58 \\ \hline \end{array}$$ Would you like further details or explanations on this? Here are five related questions to expand your understanding: 1. How can you determine the turning points of this cubic function? 2. What does the symmetry of the function's table of values tell you about the graph? 3. How would you find the roots of this cubic equation? 4. How would the graph of this function differ if the cubic term had a positive coefficient? 5. Can you determine the behavior of this function as $$x \to \infty$$ and $$x \to -\infty$$? **Tip:** When working with cubic functions, check for symmetry or repeated values in the table, as it can help identify patterns in the graph's behavior.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Functions
Polynomial Functions

Formulas

Cubic equation formula: y = ax^3 + bx^2 + cx + d

Theorems

Polynomial evaluation at specific points

Suitable Grade Level

Grades 10-12

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