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# Writing Linear Functions

## Writing Linear Functions 1.1 - Solution

a

We want to find the rate of change for the linear function in the following diagram.

We can see that the graph passes through the points $(1,4)$ and $(2,6).$ Let's mark these points in the diagram.

Let's draw arrows to show what it looks like to move from one point to the other by taking one horizontal step followed by a vertical. The horizontal step we label $\Delta x$ and the vertical $\Delta y.$

In the diagram we can see that $\Delta x=1$ and $\Delta y=2.$ By using these values in the relationship $\text{rate of change}=\dfrac{\Delta y}{\Delta x}$ we find the rate of change.
$\text{rate of change}=\dfrac{\Delta y}{\Delta x}$
$\text{rate of change}=\dfrac{{\color{#009600}{2}}}{{\color{#0000FF}{1}}}$
$\text{rate of change}=2$
Thus, the rate of change for the graphed function is $2.$
b

Observing the given graph, we can see that the line passes through the points $(0,6)$ and $(1.5,0).$ We can move from one point to the other by taking one horizontal step, $\Delta x,$ followed by a vertical step, $\Delta y.$

We can in the graph see that the horizontal step we take is $1.5,$ making $\Delta x=1.5.$ The vertical step is down $6$ units. Moving down makes the step negative. Therefore, $\Delta y=\text{-} 6.$ We find the rate of change using this relationship. $\text{rate of change}=\dfrac{\Delta y}{\Delta x}$ Let's use the values we have found to calculate the rate of change.
$\text{rate of change}=\dfrac{\Delta y}{\Delta x}$
$\text{rate of change}=\dfrac{{\color{#009600}{\text{-} 6}}}{{\color{#0000FF}{1.5}}}$
$\text{rate of change}=\text{-} 4$
We have found that the rate of change for the linear function is $\text{-} 4.$