Section 1.6 Linear Functions
Subsection 1. Slope-Intercept Form
Because the \(y\)-intercept \((0,b)\) is the "starting value" of a linear model, and its rate of change is measured by its slope,\(m\text{,}\) the equation for a linear model
can be expressed in symbols as
Slope-Intercept Form.
If we write the equation of a linear function in the form,
then \(m\) is the slope of the line, and \(b\) is the \(y\)-intercept.
Subsubsection Examples
Example 1.79.
The temperature inside a pottery drying oven starts at 70 degrees and is rising at a rate of 0.5 degrees per minute. Write a function for the temperature, \(H\text{,}\) inside the oven after \(t\) minutes.
At \(t=0\text{,}\) the temperature is 70 degrees, so \(b=70\text{.}\)
The slope is given by the rate of increase of \(H\text{,}\) so \(m=0.5\text{.}\)
Thus, the function is
Example 1.80.
A perfect score on a driving test is 120 points, and you lose 4 points for each wrong answer. Write a function for your score, \(S\text{,}\) if you give \(n\) wrong answers.
If \(n=0\text{,}\) your score is 120, so \(b=120\text{.}\)
Your score decreases by 4 points per wrong answer, so \(m=\dfrac{\Delta S}{\Delta n} = 4\text{.}\)
The function is
Subsubsection Exercises
Checkpoint 1.81.
Monica has saved $7800 to live on while she attends college. She spends $600 a month. Write a function for the amount, \(S\text{,}\) in Monica's savings account after \(t\) months.
Checkpoint 1.82.
Jesse opened a new doughnut shop in an old store-front. He invested $2400 in remodeling and set-up, and he makes about $400 per week from the business. Write a function giving the shop's financial standing, \(F\text{,}\) after \(w\) weeks.
Subsection 2. Point-Slope Form
If we don't know the \(y\)-intercept of a line but we do know one other point and the slope, we can still find an equation for the line.
Point-Slope Formula.
To find an equation for the line of slope \(m\) passing through the point \((x_1,y_1)\text{,}\) use the point-slope formula
or
Subsubsection Example
Example 1.83.
Find an equation for the line that passes through \((1,3)\) and has slope \(-2\text{.}\)
We substitute \(x_1=\alert{1}\text{,}\) \(y_1=\alert{3}\text{,}\) and \(m=\alert{-2}\) into the point-slope formula.
Example 1.84.
Find an equation for the line of slope \(\dfrac{-1}{2}\) that passes through \((-3,-2)\text{.}\)
We substitute \(x_1=\alert{-3}\text{,}\) \(y_1=\alert{-2}\text{,}\) and \(m=\alert{\dfrac{-1}{2}}\) into the point-slope formula.
Subsubsection Exercises
Checkpoint 1.85.
Find an equation for the line of slope \(-4\) that passes through \((2,-5)\text{.}\)
Checkpoint 1.86.
Find an equation for the line of slope \(\dfrac{2}{3}\) that passes through \((-6,1)\text{.}\)
Subsection 3. Graphing a line
If we know one point on a line and its slope, we can sketch its graph without having to make a table of values.
Subsubsection Examples
Example 1.87.
Graph the line \(~y=\dfrac{3}{4}x-2\)
Step1: Begin by plotting the \(y\)-intercept, \((0,-2)\text{.}\)
Step 2: We use the slope, \(\dfrac{\Delta y}{\Delta x} = \dfrac{3}{4}\text{,}\) to find another point on the line, as follows. Start at the point \((0,-2)\) and move 3 units up and 4 units to the right. Plot a second point here, at \((4,1)\text{.}\)
Step 3: Find a third point by writing the slope as \(\dfrac{\Delta y}{\Delta x} = \dfrac{-3}{-4}\text{:}\) from \((0,-2)\text{,}\) move down 3 units and 4 units to the left. Plot a third point here, at \((-4,-5)\text{.}\)
Finally, draw a line through the three points.
Example 1.88.
Graph the line of slope \(\dfrac{-1}{2}\) that passes through \((-3,-2)\text{.}\)
Step 1: Begin by plotting the point \((-3,-2)\text{.}\)
Step 2: Use the slope, \(\dfrac{\Delta y}{\Delta x} = \dfrac{-1}{2}\text{,}\) to find another point on the line, as follows. Start at the point \((-3,-2)\) and move 1 unit down and 2 units to the right. Plot a second point here, at \((-1,-3)\text{.}\)
Step 3: Find a third point by writing the slope as \(\dfrac{\Delta y}{\Delta x} = \dfrac{1}{-2}\text{:}\) from \((-3,-2)\text{,}\) move 1 unit up and 2 units to the left. Plot a third point here, at \((-5,-1)\text{.}\)
Finally, draw a line through the three points.
Subsubsection Exercises
Checkpoint 1.89.
Graph the line \(~y=\dfrac{-1}{3}x-3\)
Checkpoint 1.90.
Graph the line with slope \(m=\dfrac{3}{2}\) passing through \((-1,-2)\text{.}\)