Question

What is the rate of change and initial value for the linear relation that includes the points shown in the table?

x y
1 20
3 10
5 0
7 −10

Answer 1

Each time x increases by 2 (eg from x = 1 to x = 3), the value of y drops by 10 (eg from y = 20 to y = 10)

Therefore the slope is...
slope = rise/run = (change in y)/(change in x) = -10/2 = -5
slope = -5

So each time x increases by 1, y will decrease by 5
Flip things around: each time x decreases by 1, y will increase by 5

So the pair (x,y) = (1,20) shown in row 1 of the table leads to (0,25) based on the rule above. Another way to see this is to plug m = -5, x = 1 and y = 20 into y = mx+b and solve for b to get b = 25

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Answers:
Rate of change = -5
Initial value = 25

Notes: slope is another term for rate of change. The y intercept is another way of stating the initial value

Answer 2

The function of the linear relation is y=-5x+25.

Given to us,

x         y

1         20

3         10

5         0

7         10

From row 1,

$y=mx+c\\(20) = m(1)+c\\20=m+c$

therefore, equation 1, 20=m+c

From row 3,

$y=mx+c\\0=m(5) + c\\c= -5m$

therefore, equation 2, c= -5m.

Substituting the value of c in equation 1,

$20 = m+c\\20= m+(-5m)\\20= m -5m\\20=-4m\\m = \dfrac{20}{-4}\\m=-5$

Therefore, the slope of function or rate of change for the linear relation is -5.

substituting the value of m in c,

$c=-5m\\c=-5(-5)\\c = 25$

Hence, the function of the linear relation is y=-5x+25.

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