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Writing Linear Functions

Writing Linear Functions 1.10 - Solution

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a

Let's start by recalling the slope-intercept form of a line. y=mx+b\begin{gathered} y=m x+b \end{gathered} Here, mm is the slope and bb the y-y\text{-}intercept. We'll find these two values for the given line.

Finding the y-y\text{-}intercept

Consider the given graph.

The value of bb is given by the y-y\text{-}coordinate of the point at which the line intercepts the y-y\text{-}axis. We can see in the graph that the line intercepts the y-y\text{-}axis at (0,2).(0,2). This means that b=2.b=2. y=mx+2\begin{gathered} y=m x+2 \end{gathered}

Finding the slope

To find the slope, we will trace along the line on the given graph until we find a lattice point, which is a point that lies perfectly on the grid lines. By doing this, we will be able to identify the slope using the rise and run of the graph.

Here we have identified (1,1)(1,1) as our second point. Traveling to this point from the y-y\text{-}intercept requires 11 step down and 11 step to the right. riserun=-11m=-1\begin{gathered} \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{-}1}{1} \quad\Leftrightarrow\quad m= \text{-}1 \end{gathered} We can now write the complete equation of the line. y=-x+2\begin{aligned} y=\text{-} x +2 \end{aligned}

b
To find the equation of the line we must identify the slope and the yy-intercept.

Finding the y-y\text{-}intercept

Consider the given graph.

Thus, the yy-intercept is b=-1b=\text{-}1 and we can substitute it into the slope-intercept form. y=mx1\begin{gathered} y=mx-1 \end{gathered}

Finding the Slope

The slope can be found by identifying the rise and run between two points on the line. We already know one point, the yy-intercept. Let's now use a lattice point, a point that lies perfectly on the grid, as our second point.

We have identified (2,0)(2,0) as our second point. Traveling to this point from the y-y\text{-}intercept requires 22 steps to the right and 11 steps up. riserun=12m=12\begin{gathered} \dfrac{\text{rise}}{\text{run}} = \dfrac{1}{2} \quad\Leftrightarrow\quad m = \dfrac{1}{2} \end{gathered} We can now write the complete equation of the line. y=12x1\begin{aligned} y=\dfrac{1}{2}x-1 \end{aligned}