Problem 19 - Contemporary Precalculus

Question

Each equation defines y as a function of \(x .\) Create a table that shows the values of the function for the given values of \(x\) $$y=x^{2}+x-4 ; \quad x=-2,-1.5,-1, \ldots, 3,3.5,4$$

Step-by-Step Solution

Answer

Question: Create a table of values for the function \(y=x^2+x-4\) for the given set of x-values from -2 to 4 with a step of 0.5. Answer: | x | y | |-------|--------| | -2 | -2 | | -1.5 | -3.25 | | -1 | -4 | | -0.5 | -3.75 | | 0 | -4 | | 0.5 | -3.25 | | 1 | -2 | | 1.5 | -0.25 | | 2 | 2 | | 2.5 | 4.25 | | 3 | 8 | | 3.5 | 12.25 | | 4 | 16 |

Step 1: Identify the given function and x-values

The given function is \(y=x^2+x-4\). We will be evaluating this function at the given x-values: $$x = -2, -1.5, -1, \ldots, 3, 3.5, 4.$$

Step 2: Calculate y-values

For each x-value, substitute the x-value into the function's equation and compute the corresponding y-value as follows: $$ y(x) = x^2+x-4 $$ For x = -2: $$ y = (-2)^2+(-2)-4 = 4-2-4 = -2 $$ For x = -1.5: $$ y = (-1.5)^2+(-1.5)-4 = 2.25-1.5-4 = -3.25 $$ For x = -1: $$ y = (-1)^2+(-1)-4 = 1-1-4 = -4 $$ Continue this process for the remaining x-values in the given range (-0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4).

Step 3: Organize the values in a table

Now, we will organize the computed y-values with their corresponding x-values in a table: | x | y | |-------|--------| | -2 | -2 | | -1.5 | -3.25 | | -1 | -4 | | ... | ... | | 3 | y(3) | | 3.5 | y(3.5) | | 4 | y(4) | Complete the table by calculating the y-values for the remaining x-values using the same process as in Step 2.