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Section 1.5 Slope

Subsection 1. Use ratios for comparison

Slope is a type of ratio that compares vertical distance per unit of horizontal distance. We use ratios for comparison in other situations, for example, when shopping we might compute price per unit.

Subsubsection Examples

Example 1.67.

You are choosing between two brands of iced tea. Which is a better bargain: a 28-ounce bottle of Teatime for $1.82, or a 36-ounce bottle of Leafdream for $2.25?

Solution

Compute the ratio price per ounce for each brand.

\begin{equation*} \text{Teatime:}~~~\dfrac{182 \text{ cents}}{28 \text{ ounces}}= 6.5 \text{ cents per ounce} \end{equation*}
\begin{equation*} \text{Leafdream:}~~~\dfrac{225 \text{ cents}}{36 \text{ ounces}}= 6.25 \text{ cents per ounce} \end{equation*}

Leafdream is the better bargain.

Example 1.68.

The trail to Lookout Point gains 780 feet in elevation over a distance of 1.3 miles. The trail to Knife Edge gains 950 feet in elevation over a distance of 1.6 miles. Which trail is steeper?

Solution

Compute the ratio of elevation gain to horizontal distance traveled for each trail.

\begin{equation*} \text{Lookout Point:}~~~\dfrac{780 \text{ feet}}{1.3 \text{ miles}}= 600 \text{ feet per mile} \end{equation*}
\begin{equation*} \text{Knife Edge:}~~~\dfrac{950 \text{ feet}}{1.6 \text{ miles}}= 593.75 \text{ feet per mile} \end{equation*}

The Lookout Point trail is steeper.

Subsubsection Exercises

Rachel drove 292.4 miles on 8.6 gallons of gasoline. Reuben drove 390 miles on 12 gallons of gasoline. Who got the better gas mileage?

Hint: Compute the ratio miles per gallon.

Answer

Rachel: 34 miles per gallon; Reuben: 32.5 miles per gallon

Leslie drove 168 miles in 2.8 hours, and Mark drove 224 miles in 3.5 hours. Who drove at the greater average speed?

Hint: Compute the ratio miles per hour.

Answer

Mark: 64 miles per hour; Leslie: 60 miles per hour

Subsection 2. Calculate slope from a graph

We often think of slope as measuring the "steepness" of a graph, but the appearance of steepness is also affected by the scales on the axes.

Subsubsection Examples

Example 1.71.

Calculate the slope of the line.

line on grid
Solution

Choose two points on the line, and calculate the ratio of vertical change to horizontal change. Use the grid lines on the graph, but don't forget to note the scales on the axes.

line marked with horizontal and vertical change between two points

The slope is the ratio \(\dfrac{\Delta h}{\Delta t}\text{.}\) The variable on the horizontal axis increases by 4 units, from 2 to 6, so \(\Delta t=4\text{.}\) The variable on the vertical axis increases by 8 grid lines, but each grid line represents 2 units, so \(\Delta h=16\text{.}\) Thus, the slope is \(\dfrac{\Delta h}{\Delta t} = \dfrac{16}{4}=4\text{.}\)

Example 1.72.

Calculate the slope of the line.

line on grid
Solution

Choose two points on the line, and calculate the ratio of vertical change to horizontal change. Use the grid lines on the graph, but don't forget to note the scales on the axes.

line marked with horizontal and vertical change between two points

The slope is the ratio \(\dfrac{\Delta V}{\Delta t}\text{.}\) The horizontal variable, \(t\text{,}\) increases by 6 grid lines, but each grid line represents 2 units, so \(\Delta t=12\text{.}\) The vertical variable, \(V\text{,}\) decreases by 3 grid lines, or 6 units, so \(\Delta V=-6\text{.}\) Thus, \(\dfrac{\Delta V}{\Delta t} = \dfrac{-6}{12}=\dfrac{-1}{2}\text{.}\)

Subsubsection Exercises

Calculate the slope of the line.

Hint: Find two points that lie on the intersection of grid lines, so that it's easy to read their coordinates. For example, you could use \((2, 300)\) and \((8, 600)\text{.}\)

line on grid
Answer
\(50\)

Calculate the slope of the line.

Hint: Find two points that lie on the intersection of grid lines. For example, you could use \((0, 60)\) and \((3, -12)\text{.}\)

line on grid
Answer
\(-24\)

Subsection 3. Calculate slope using a formula

Recall that the subscripts on the coordinates in \(~P_1(x_1,y_1)~\) and \(~P_2(x_2,y_2)~\) just mean "first point" and "second point".

Two-Point Formula for Slope.

The slope of the line joining points \(~P_1(x_1,y_1)~\) and \(~P_2(x_2,y_2)~\) is

\begin{equation*} m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2-y_1}{x_2-x_1}~~~~~~\text{if}~~~~x_2 \not= x_1 \end{equation*}

Subsubsection Example

Example 1.75.

Compute the slope of the line joining \((-6,2)\) and \((3,-1)\text{.}\)

Solution

It doesn't matter which point is \(P_1\) and which is \(P_2\text{,}\) so we choose \(P_1\) to be \((-6,2)\text{.}\) Then \((x_1,y_1)=(-6,2)\) and \((x_2,y_2)=(3,-1)\text{.}\) Thus,

\begin{align*} m \amp = \dfrac{y_2-y_1}{x_2-x_1}\\ \amp = \dfrac{-1-2}{3-(-6)} = \dfrac{-3}{9} = \dfrac{-1}{3} \end{align*}
Caution 1.76.

Make sure that you subtract both the \(x\) and \(y\) coordinates in the same order! That is, do NOT calculate

\begin{equation*} \dfrac{y_{\alert{2}}-y_{\alert{1}}}{x_{\alert{1}}-x_{\alert{2}}}~~~~~~~~~~\blert{\text{Incorrect!}} \end{equation*}

or your slope will have the wrong sign.

Subsubsection Exercises

Compute the slope of the line joining the points \((5,2)\) and \((8,7)\text{.}\)

Answer
\(\dfrac{5}{3}\)

Compute the slope of the line joining the points \((-3,-4)\) and \((-7,1)\text{.}\)

Answer
\(\dfrac{-5}{4}\)