Answer:
Slope (rate of change) of the linear function: [tex](-2/3)[/tex].
Value of the function would be [tex]22[/tex] when [tex]x = 24[/tex].
Step-by-step explanation:
To find the rate of change (slope) of a linear function, divide the change in the output (rise) by the change in the input (run). If a linear function goes through points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], the slope of the function would be:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}}\end{aligned}[/tex].
Given the two points on the graph of this function, the slope of the function in this question would be:
[tex]\begin{aligned}m &= \frac{26 - 34}{18 - 6} = -\frac{2}{3}\end{aligned}[/tex].
To extrapolate and find the function value for a different input value, start by finding the equation for this function. If the slope of a linear function is [tex]m[/tex] and the graph of the function goes through the point [tex](x_{0},\, y_{0})[/tex], the equation of the function would be:
[tex]y - y_{0} = m\, (x - x_{0})[/tex].
Rearrange to find an expression for the function output [tex]y[/tex]:
[tex]y = m\, (x - x_{0}) + y_{0}[/tex].
In this question, [tex]m = (-2/3)[/tex]. Using the point [tex](18,\, 26)[/tex], the equation for this function would be:
[tex]\displaystyle y = \left(-\frac{2}{3}\right)\, (x- 18) + 26[/tex].
Substitute in [tex]x = 24[/tex] and evaluate to find the value of the function:
[tex]\begin{aligned} y = \left(-\frac{2}{3}\right)\, (24 - 18) + 26 = 22\end{aligned}[/tex].
