Unit 2 is about functions that change at a constant rate, which is why their graphs are straight lines. By the end, you should be able to write a line's equation when you have enough information, such as two distinct points, graph it, model a real situation with it, and fit a line to messy data.
Read the overview first, then work through each topic's notes and try the checkpoint question before moving on.
Linear functions are the simplest functions you'll study in pre-calculus, and they set up ideas you'll reuse all year. Slope is the first version of "rate of change," which becomes the central idea of calculus. This unit moves from pure algebra to modeling real data.
Honors Pre-Calculus Unit 2 covers linear functions of the form . You calculate slope from two points, write equations in slope-intercept, point-slope, and standard form, and graph lines using intercepts and slope. You identify parallel and perpendicular lines using their slopes, including the special case of vertical and horizontal lines, and you solve systems of two linear equations. The second half applies all of this to modeling. You build linear models from word problems, interpret slope and intercept in context, choose a sensible domain, and fit a line of best fit to scatter plot data, then use residuals and the correlation coefficient to assess the model.
The slope gives the same value for any two points with different x-values. That constant slope is what makes the graph a straight line. If a table gives you different slopes for different pairs, the data is not linear.
In a model like for a taxi fare, the slope 3 is the cost per mile and the intercept 5 is the starting fee. When you interpret these numbers, include their units. The domain should also match the situation, since you can't ride negative miles.
A line of best fit summarizes the trend in a scatter plot. Residuals show how far each point sits from the line. The correlation coefficient tells you how strong the linear pattern is.
Writing equations, graphing, checking for parallel or perpendicular lines, and building models all depend on finding and reading the slope. When you fit a line to data, you're estimating the rate of change that best describes the trend. Later units will study functions whose rate of change is not constant, so this is the baseline you'll compare everything to.
A linear function has the form f(x) = mx + b.
To graph y = mx + b, plot the y-intercept (0, b), then use the slope as rise over run to find a second point.
A linear model turns a word problem into y = mx + b.
Real data rarely falls on a perfect line, so you start with a scatter plot.
Open this guide for a closer review of the topic.
Open this guide for a closer review of the topic.
A linear function has the form . The slope tells you how much the output changes when the input goes up by 1. The y-intercept is the output when .
For two points with different x-values, you find slope with . A positive slope means the function increases, a negative slope means it decreases, and a slope of 0 means it stays constant.
The form you write depends on what you know. If you know the slope and intercept, use slope-intercept form . If you know a point and the slope, use point-slope form . Standard form is useful for finding intercepts quickly.
Distinct nonvertical parallel lines have equal slopes. When both perpendicular lines have defined slopes, their slopes are negative reciprocals, so their product is . For example, a line with slope is perpendicular to a line with slope . Vertical lines are parallel to each other and perpendicular to horizontal lines.
Worked example. Write the equation of the line through and . The slope is . Point-slope form gives . Distribute to get , so . Check with the other point. , which matches.
Check yourself: What is the slope of a line perpendicular to ? (Answer: , because the negative reciprocal of is , and .)
| Form | Equation | Use it when |
|---|---|---|
| Slope-intercept | $y = mx + b$ | You know the slope and y-intercept, or you want to graph quickly |
| Point-slope | $y - y_1 = m(x - x_1)$ | You know one point and the slope, or two points |
| Standard | $Ax + By = C$ | You want both intercepts fast or are setting up a system for elimination |
To graph , plot the y-intercept , then use the slope as rise over run to find a second point. For , move right 2 and up 3. Draw the line through both points.
The x-intercept is where . For , setting gives , and setting gives . Plot and and connect them.
Two special lines show up often. A horizontal line like has slope 0 and is a constant function. A vertical line like has undefined slope, because the run is 0 and you can't divide by 0. A vertical line fails the vertical line test, so it is not a function.
A system of two linear equations asks where two lines meet. The lines can cross once (one solution), be parallel (no solution), or be the same line (infinitely many solutions). You can solve by graphing, substitution, or elimination.
Worked example. Solve and . Both equal , so set them equal. gives , so . Then . The lines cross at .
Check yourself: How many solutions does the system and have? (Answer: None. The slopes are equal and the intercepts are different, so the lines are parallel and never meet.)
| What the lines look like | Slopes and intercepts for nonvertical lines | Number of solutions |
|---|---|---|
| Cross at one point | Different slopes | One |
| Parallel | Same slope, different intercepts | None |
| Same line | Same slope, same intercept | Infinitely many |
A linear model turns a word problem into . Start by naming the independent variable (the input, like time or miles) and the dependent variable (the output, like cost). The slope is the rate, usually a "per" quantity such as dollars per mile. The intercept is the starting amount when the input is 0.
Interpret every number with units. In , the slope means each mile adds 3 dollars, and the intercept means the fare starts at 5 dollars before you go anywhere.
Pick a domain that fits the situation. Miles can't be negative, so , and there may be an upper limit too. The range is the set of outputs produced by the allowed inputs. For a closed, bounded interval, you can use its endpoints to find the range of a linear model.
Predictions within the range of your known data are interpolation, but they can still be unreliable if the model fits poorly or the data has large gaps. Predictions outside that range are extrapolation and can fail, because real rates rarely stay constant forever.
Worked example. A taxi ride of 2 miles costs 11 dollars and a ride of 5 miles costs 20 dollars. The slope is dollars per mile. Substitute into to get , so . The model is . With 35 dollars, solve to get , so you can ride 10 miles.
Check yourself: A gym charges a 40-dollar sign-up fee plus 25 dollars per month. What linear model gives the total cost after months, and what does the slope mean? (Answer: . The slope 25 means each additional month adds 25 dollars.)
Real data rarely falls on a perfect line, so you start with a scatter plot. Look for direction (positive, negative, or none), strength (tight or spread out), form (straight or curved), and outliers. If the points show a clear curve, consider a nonlinear model.
The line of best fit, also called the least squares regression line, is the line that makes the sum of the squared residuals as small as possible. A calculator finds it for you. Your job is to interpret the slope and intercept in context, using phrases like "predicted" or "on average."
A residual is actual minus predicted, . A positive residual means the point is above the line. Random scatter in a residual plot supports a linear model, but it does not guarantee accurate predictions. A curved pattern suggests that a linear model misses part of the relationship.
The correlation coefficient runs from to and measures how strong the linear pattern is. Values near are strong. Its square, , tells you the fraction of the variation in that the line explains.
Worked example. A regression based on study times from 0 to 8 hours gives the predicted test score . Someone who studied 4 hours scored 75. The prediction is , so the residual is . That score is 6.2 points above the line. Predicting 12 hours gives 110.4, which is impossible on a 100-point test, showing why extrapolation is risky.
Check yourself: A line of best fit predicts 52 for a data point whose actual value is 47. What is the residual, and is the point above or below the line? (Answer: . It is negative, so the point is 5 units below the line.)
| Term | Definition |
|---|---|
| Linear function | A function of the form $f(x) = mx + b$ that has a constant rate of change and a straight-line graph. |
| Slope | The rate of change of a nonvertical line, calculated as $\frac{y_2 - y_1}{x_2 - x_1}$ using two points with different x-values. A vertical line has undefined slope. |
| y-intercept | The value of $y$ when $x = 0$. The corresponding point $(0, y)$ is where the line intersects the y-axis. |
| Slope-intercept form | The equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. |
| Parallel lines | Distinct lines in the same plane that never intersect. Nonvertical parallel lines have equal slopes; vertical lines are also parallel to each other. |
| Perpendicular lines | Lines that meet at a right angle. If both slopes are defined, their product is $-1$. A vertical line and a horizontal line are also perpendicular. |
| System of linear equations | Two or more linear equations solved together to find the point or points they share. |
| Line of best fit | The line that minimizes the sum of squared residuals for a set of data points. |
| Residual | The difference between an actual data value and the value the model predicts. |
| Interpolation | Using a model to predict values within the range of the data it was built from. |
Computing flips the sign of your slope. Whichever point you start with in the numerator, start with the same point in the denominator. Labeling the points as and before plugging in prevents this.
Flipping to gives a line that is not perpendicular. You need to flip the fraction and change the sign, so the perpendicular slope is . Check by multiplying the slopes, which should give .
A horizontal line like has slope 0 because there is no rise. A vertical line like has undefined slope because there is no run, and dividing by 0 is impossible. Picture which one you could walk along without climbing.
A model only describes the trend it was built from. Predicting well beyond the data can produce impossible results, like a test score over 100. A strong correlation also doesn't prove that the input causes the output, so describe the relationship as an association.