Answer:
Rate of change: m = -2/3
Value of the function at x = 24: y = 22
Step-by-step explanation:
You can write linear functions in slope-intercept form: y = mx + b
"x" and "y" are points on the line
"m" is the slope, how steep the line is, also known as rate of change
"b" is the y-intercept, when the graph hits the y-axis
Find the rate of change of the function by finding the slope between the two given points.
The formula for slope: [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
Label point 1 and point 2. Label each coordinate; remembered ordered pairs are (x, y).
Point 1 (6, 34) x₁= 6 y₁ = 34
Point 2 (18, 26) x₂ = 18 y₂ = 26
Substitute the points into your equation using the bolded information.
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
[tex]=\frac{26-34}{18-6}[/tex] Subtract
[tex]=\frac{-8}{12}[/tex] Reduce the fraction. Top and bottom divide by 4
[tex]=\frac{-2}3}[/tex] Slope of the line, "m"
Substitute the slope and any point you want into slope-intercept form. I choose point (6, 34). x = 6 y = 34 m = -2/3
y = mx + b
34 = (-2/3)(6) + b Multiply to simplify
34 = (-12/3) + b
Start isolating "b"
34 + 12/3 = (-12/3) + 12/3 + b Add 12/3 on both sides
34 + 12/3 = b
Add fractions by combining the whole number into the numerator. Multiply the whole number by the denominator and add to the numerator.
b = (34*3 + 12) / 3 Solve numerator
b = 114/3 Value of "b"
Substitute the values "m" and "b" into slope-intercept form.
y = -2/3 x + 114/3 Equation of the line
To find the x = 24, take the equation of the line and substitute "x" for 24. Then solve for "y".
y = -2/3 x + 114/3 Equation of the line
y = (-2/3)(24) + 114/3
y = (-2*24)/3 + 114/3 Multiply into the numerator
y = (-48)/3 + 114/3 Combine under the same denominator
y = (-48 + 114) / 3 Add
y = 66/3 Divide
y = 22 Value of function
Therefore the rate of change, or slope "m", is -2/3. When "x" is 24, the value of the entire function is 22.
