Study Guide: 8-5-6. Point-Slope Form
Vocabulary
| Term | Description |
|---|---|
| Cartesian Plane | a two-dimensional plane divided into four quadrants using x- and y-axis |
| Origin | the point of intersection of the x-axis and y-axis on a Cartesian Plane |
| Function | a relationship between variables that has one output for each and every input |
| Linear Function | a function that is represented by a line when graphed on a Cartesian Plane |
| Domain | the set of input values or x-values of a function |
| Range | the set of output values or y-values of a function |
| Slope | a ratio of the rate at which the dependent variable is changing versus the rate at which the independent variable is changing; frequently expressed as $\frac{RISE}{RUN}$, or $\frac{\textit{The change in y}}{\textit{The change in x}}$ |
| Slope-Intercept form | the form $y = mx + b$ of a linear equation, where m represents the slope of the line and b represents its y-intercept |
| Point-Slope form | the form $y - y_1 = m(x - x_1)$ of a linear equation, where m is the slope, and $y_1$ and $x_1$ are the coordinates of a point. |
| x-intercept | the point on the x-axis where the line of a function crosses the x-axis |
| y-intercept | the point on the y-axis where the line a function crosses the y-axis |
| Absolute Value | the distance a number is from zero on a number line (distance is never negative) |
Using Point-Slope Form
If you know at least one point and the slope of a line, you can use the Point-Slope Form to create the equation.
Example Problem
The cost of Internet access at a cafe is a function of time. The cost for 8, 25 and 40 minutes are shown. Write an equation in Slope-Intercept Form that represents the function. Then find the cost of surfing the web a the cafe for one hour. (Practice B, Lesson 5-7, #9)
| Time (min) | Cost ($) |
|---|---|
| 8 | 4.36 |
| 25 | 7.25 |
| 40 | 9.80 |
Steps for Solving
1. Use the Slope Formula to find the slope
\begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} \\[2ex] &= \frac{9.80 - 7.25}{40 - 25} \\[2ex] &= \frac{2.55}{15} \\[2ex] &= 0.17 \end{aligned}
2. Use the Point-Slope Form to find the equation and convert it into Slope-Intercept Form.
\begin{aligned} y - y_2 &= m(x - x_2) &&\textit{Point-Slope Form} \\[2ex] y - 9.80 &= 0.17(x - 40) &&\textit{$y_2$ and $x_2$ replaced with values} \\[2ex] y &= 0.17x + 3 &&\textit{Simplified to Slope-Intercept Form} \\\ \end{aligned}
3) Check by plugging in the coordinates for another point and simplifing. If the equation is true, it is correct.
\begin{aligned} y &= 0.17x + 3 &&\textit{The equation} \\[2ex] 7.25 &= 0.17(25) + 3 &&\textit{Using the point $(25, 7.25)$} \\[2ex] 7.25 &= 4.25 + 3 \\[2ex] 7.25 &= 7.25 &&\textit{Checked} \\\ \end{aligned}
4. To find the cost for one hour, replace $x$ (time) with 60 (minutes), and solve for $y$ (cost).
\begin{aligned} y &= mx + b &&\text{Slope-Intercept Form} \\[2ex] y &= 0.17(60) + 3 &&\textit{Replace $x$ with 60.} \\[2ex] y &= 13.2 &&\textit{Simplified} \\[2ex] y &= 13.2 &&\textit{Solution: Cost for one hour is $13.20.} \end{aligned}