Question

8 tell whether the information given is enough to write an equation for a linear function.
a. the initial value and the rate of change of the function

b. the slope of the line and the rate of change of the function

c. the slope of the line and one point on the line that is not
the ( y )-intercept

d. the ( y )-intercept of the line and the value of the function at ( x = 5 )

e. the ( y )-intercept of the line and the value of the function at ( x = 0 )

Explanation:

Step1: Analyze part a

A linear function is $y=mx+b$, where initial value = $b$ (y-intercept), rate of change = $m$ (slope). Both values are given, so we can write the equation.

Step2: Analyze part b

Slope is the rate of change for a linear function. This only gives one unique value, not two distinct pieces of information needed for $y=mx+b$.

Step3: Analyze part c

With slope $m$ and a point $(x_1,y_1)$ (not y-intercept), we use point-slope form $y-y_1=m(x-x_1)$, which can be rearranged to slope-intercept form. We have enough info.

Step4: Analyze part d

Y-intercept gives $b$ (the value at $x=0$). The value at $x=5$ gives a point $(5, y_5)$. We can calculate slope $m=\frac{y_5 - b}{5-0}$, then write $y=mx+b$. Enough info.

Step5: Analyze part e

The y-intercept is the value of the function at $x=0$. This only gives one unique value ($b$), no slope information. We cannot write the full linear equation.

Answer:

a. Yes
b. No
c. Yes
d. Yes
e. No