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Algebra 1 Unit 3: Linear functions & graphs

A linear function has a constant rate of change, so its graph is a straight line. Three forms of the equation each make something easy to see.

Covers: Slope, slope-intercept, point-slope and standard form, intercepts, parallel and perpendicular lines

Study guide

Slope measures steepness: rise over run, m = (y₂ − y₁)/(x₂ − x₁). Positive slopes rise left to right, negative slopes fall, horizontal lines have slope 0, and vertical lines have undefined slope. In real situations slope is a rate of change, like dollars per hour.

Linear equations come in three common forms. Slope-intercept form y = mx + b shows the slope and y-intercept, so you can graph by plotting b and using the slope to find more points. Point-slope form y − y₁ = m(x − x₁) is best when you know a point and the slope, or two points. Standard form Ax + By = C is handy for finding intercepts: set x = 0 for the y-intercept and y = 0 for the x-intercept.

The x-intercept is where a line crosses the x-axis (y = 0) and the y-intercept is where it crosses the y-axis (x = 0). In context, the y-intercept is usually a starting value.

Parallel lines have the same slope and never meet. Perpendicular lines meet at right angles, and their slopes are negative reciprocals: a line with slope 3 is perpendicular to one with slope −1/3. To write the equation of a parallel or perpendicular line through a point, find the needed slope and use point-slope form.

A relation is a function if each input has exactly one output. In a list of ordered pairs, no x-value can repeat with a different y-value, and on a graph every vertical line meets the graph at most once (the vertical line test). f(x) names the output of function f for input x, so f(4) means substitute 4 for x. The domain is the set of allowed inputs and the range is the set of outputs; in context the domain is often limited to sensible values, such as whole numbers of items.

The average rate of change of f from x = a to x = b is [f(b) − f(a)]/(b − a), the slope of the line through those two points. For a nonlinear function it changes from one interval to another, so always use the interval you are given. A function is increasing on an interval if its graph rises from left to right there and decreasing if it falls. Read intervals using x-values, and compare functions by checking which ones rise across the whole interval in question.

The graph of y = |x| is a V shape with its vertex at the origin. The graph of y = |x − h| + k is the same V moved right h and up k, with its vertex at (h, k). In general, f(x) + k shifts a graph up k, and f(x − h) shifts it right h; with f(x + h), the shift is to the left.

Key ideas

• Slope-intercept y = mx + b shows slope m and y-intercept b.
• Point-slope y − y₁ = m(x − x₁) is fastest when you know a point and slope.
• Standard form Ax + By = C makes intercepts easy: set x or y to 0.
• Horizontal lines have slope 0; vertical lines have undefined slope.
• Parallel: equal slopes. Perpendicular: slopes multiply to −1.

Formulas and rules

Common mistakes

• Mixing up which point's coordinates go on top.
• Using the negative but not the reciprocal for a perpendicular slope.

Words to know

Slope-intercept form

y = mx + b

Point-slope form

y − y₁ = m(x − x₁)

Slope of a horizontal line

0

Perpendicular slopes

Negative reciprocals: slope 3 → slope −1/3.

X-intercept

Where a graph crosses the x-axis; set y = 0 and solve.

Slope

Rise over run: (y₂ − y₁)/(x₂ − x₁).

Y-intercept

Where a graph crosses the y-axis (x = 0).

Standard form (line)

Ax + By = C

Undefined slope

The slope of a vertical line.

Parallel lines

Same slope, different intercepts.

Rate of change

How much the output changes per unit of input.

Function

A rule giving exactly one output for each input.

Sample practice questions

39 more questions like these, from level 1 to 5, are in the practice set.