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Chapter 1: Problem 2

A table of values for a linear function is given. Fill in the missing value and calculate \(m\) in each case. $$ \begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\ \hline y & -1 & -3 & \\ \hline \end{array} $$

Short Answer

Expert verified
)\) is \(-5\), and the slope of the linear function is \(m=-2\). The complete table is: $$ \begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\\ \hline y & -1 & -3& -5 \\\ \hline \end{array} $$

Step by step solution

01

Analyze the given table

First, let's look at the given table: $$ \begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\\ \hline y & -1 & -3 & \\\ \hline \end{array} $$ It's a linear function with two known points: \((-1,-1)\) and \((0,-3)\). We need to find the y value for the point \((1, ?)\).
02

Calculate the slope, \(m\)

To find the slope of the linear function, we need to use the slope formula, which is: $$m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the two known points \((-1,-1)\) and \((0,-3)\), we can plug in their values as: $$m = \frac{-3 - (-1)}{0 - (-1)}$$
03

Simplify the slope expression

Now, let's simplify the expression to find the value of \(m\): $$m = \frac{-3 + 1}{1} = \frac{-2}{1}$$ So, we have \(m=-2\).
04

Determine the relationship between x and y

With the slope \(m=-2\), we can write the relationship between x and y in the point-slope form: $$y - y_1 = m(x - x_1)$$ We can rewrite this equation using our slope \(m=-2\) and one of the known points such as \((-1,-1)\). Let's plug in the values: $$y - (-1) = -2(x - (-1))$$
05

Simplify the relationship equation

Now, let's simplify the equation: $$y + 1 = -2(x + 1)$$
06

Find the missing y value

To find the missing y value for the point \((1, ?)\), we will just plug in the x value into the equation and solve for y: $$y + 1 = -2(1 + 1)$$ $$y + 1 = -2(2)$$ $$y + 1 = -4$$ $$y = -4 - 1$$ $$y = -5$$ So, the missing y value for the point \((1, ?)\) is \(-5\).
07

Update the table with the missing value

Finally, let's update the table with the missing y value and the slope \(m\): $$ \begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\\ \hline y & -1 & -3& -5 \\\ \hline \end{array} $$ Slope, \(m = -2\).

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Most popular questions from this chapter

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