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Advanced Placement /AP Calculus AB: 1.2.2 Graphing Lines

AP Calculus AB: 1.2.2 Graphing Lines

Advanced Placement14 CardsCreated 7 days ago

This content covers the fundamentals of graphing linear equations using slope-intercept and point-slope forms. It explains how to calculate slope, determine y-intercepts, and write equations of lines through given points. The section also includes concepts like parallel and perpendicular lines and applies real-world problems to reinforce slope-based reasoning.

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Graphing Lines

  • A graph is a way of illustrating a set of ordered pairs. One of the easiest objects to graph is the line. Lines have direction, but no thickness.

  • The slope-intercept form, y = mx + b, and the point-slope form, (y – y1) = m(x – x1), are two means of describing lines.

  • When writing the equation of a line, the point-slope form is easier to use than the slope-intercept form because you can use any point.

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Key Terms

Term
Definition

Graphing Lines

  • A graph is a way of illustrating a set of ordered pairs. One of the easiest objects to graph is the line. Lines have direction, but no thic...

slope-intercept form

  • You can describe a line by an equation that relates the
    x-values and y-values of the points on the line. One form of the equation of a l...

note


  • Here is an example that gives two points and asks you to find the equation of the line passing through them.

  • First calcu...

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Write the equation of the line through the points (−1, 3) and (3, 1) in point-slope form.

(y-1) = -1/2 (x-3)

Joey the mountain climber is hiking up a mountain with slope 2/3. Using his altimeter he finds that he is gaining altitude at a rate of 6000 feet/hour. How fast is he hiking?

10817 feet/hour

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What is the slope of the line defined by 2x + 3y − 4 = 0?

m = −2/3

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AP Calculus AB: 1.2.2 Graphing Lines
TermDefinition

Graphing Lines

  • A graph is a way of illustrating a set of ordered pairs. One of the easiest objects to graph is the line. Lines have direction, but no thickness.

  • The slope-intercept form, y = mx + b, and the point-slope form, (y – y1) = m(x – x1), are two means of describing lines.

  • When writing the equation of a line, the point-slope form is easier to use than the slope-intercept form because you can use any point.

slope-intercept form

  • You can describe a line by an equation that relates the
    x-values and y-values of the points on the line. One form of the equation of a line is the slope-intercept form. This form makes it easy to graph a line because the y-intercept b and the slope m show up distinctly.

  • The slope is the pitch of the line. To calculate it, you will need two points on the line. Label their y-coordinates y1 and y2 , respectively, and their x-coordinates x1 and x2, respectively. Divide y2 – y1 by x2 – x1 . If you change the order of the coordinates, you must change the order of both the x-coordinates and the y-coordinates.

note


  • Here is an example that gives two points and asks you to find the equation of the line passing through them.

  • First calculate the slope. You can think of it as the difference of the y-values divided by the difference of the x-values.

  • Neither of the points you are given is the y-intercept, so you will have to calculate it. Since the line must pass through (–1, 4), you can substitute those coordinates in place of x and y. This leaves b as the only unknown.

  • After you solve for b, substitute its value and the slope into the slope-intercept form.

  • If your goal is to write the equation of a line, you may find it easier to use the point-slope form. First, use any two points to calculate the slope. Then use the coordinates of any point on the graph to arrive at the equation of the line.

Write the equation of the line through the points (−1, 3) and (3, 1) in point-slope form.

(y-1) = -1/2 (x-3)

Joey the mountain climber is hiking up a mountain with slope 2/3. Using his altimeter he finds that he is gaining altitude at a rate of 6000 feet/hour. How fast is he hiking?

10817 feet/hour

What is the slope of the line defined by 2x + 3y − 4 = 0?

m = −2/3

Find the slope of the line passing through the points (−2, 3) and (4, −5).

−4/3

What is the slope of the line described by the equation 5x − 3y + 10 = 0?

5/3

What is the y-intercept of the line defined by 2x + 3y − 4 = 0?

b = 4/3

Which of the following is the graph of the line y = −3x − 1?


  • passes through -1 and increases negatively with steep slope towards quadrant 4

  • graph 1

Which equation represents a line that is perpendicular to the given graph and passes through the origin?

y = -1/2x

What is the slope of this line segment?

m = 5/6

Write the equation of the line that passes through the point (1, −2) and is parallel to the line −6x + 3y + 48 = 0 in point-slope form.

( y + 2) = 2 (x − 1)

Find the value of a so that the slope of the line passing through the points (1, 4) and (a, a) is 1/4.

None of the above.