Chapter 1 Functions and Their Graphs. 1.1 Lines in the Plane Objectives : Find the slopes of lines....
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Transcript of Chapter 1 Functions and Their Graphs. 1.1 Lines in the Plane Objectives : Find the slopes of lines....
Pre-Calculus Chapter 1
Functions and Their Graphs
1.1 Lines in the PlaneObjectives:
Find the slopes of lines. Write linear equations given points on
lines and their slopes. Use slope-intercept forms of linear
equations to sketch lines. Use slope to identify parallel and
perpendicular lines.
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Warm Up 1.1 – Use “ZOOM SQUARE”
For each set of
equations, use your
calculator to compare
the slopes of the
lines.
What do you
observe?
Set 1 Set 2
y = 0.5x
y = –0.5x
y = x y = –x
y = 2x y = – 2x
y = 4x y = – 4x
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VocabularySlope of a LinePoint-Slope Form of a LineLinear FunctionSlope-Intercept Form a LineGeneral Form of the Equation of a LineParallel LinesPerpendicular LinesLinear ExtrapolationLinear Interpolation
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Find the Slope and Equation of the Line
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(5, 19)
(10, 6)
-4 -2 2 4 6 8 10 12 14
-5
5
10
15
20
25
30
x
y
Definition of the Slope of a Line
Slope = Change in y Change in x
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Slope of a LineDraw a diagram of each:
A line with positive slope (m > 0)
______________ from left to right.
A line with negative slope (m < 0)
______________ from left to right.
A line with zero slope (m = 0) is
________________.
A line with undefined slope is
_________________.
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Point-Slope Form of Line
Let (x1, y1) be a point on
the line whose slope is m.
If (x, y) is any other point
on the line, then
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Point-Slope Equation
The point-slope form of the equation of the
line that passes through the point (x1, y1) and
has a slope of m is
y – y1 = m (x – x1)
This equation is used frequently in
calculus.9
Example 1: Application Problem
During 2000, Nike’s net sales were $9.0
billion, and in 2001 net sales were $9.5
billion. Write a linear equation giving the net
sales y in terms of the year x. Then use the
equation to predict net sales for 2002.
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Slope-Intercept Form
The graph of the equation y = mx + b is a
line whose slope is m and whose y-
intercept is (0, b).
This can be written as a Linear Function
f (x) = mx + b.
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Example 2 Determine the slope and y-intercept of each
linear equation. Then describe the graph of
each equation.
a. 3x + 4y = 1
b. y = 12
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General Form of Linear EquationA horizontal line (m = 0) has an equation
of the form y = b.
A vertical line has an equation of the form
x = a. Can this equation be written in slope-
intercept form? Why?
Any line can be written in General Form:
Ax + By + C = 0 or Ax + By = c13
Summary of Equations of Lines
1. General Form Ax + By + C = 0
2. Vertical Line x = a
3. Horizontal Line y = b
4. Slope-Intercept y = mx + b
5. Point-Slope y – y1 = m(x – x1)
14
Parallel & Perpendicular LinesParallel Lines
Never intersect.
Slopes are equal: m1 = m2
Perpendicular Lines
Intersect at a right angle.
Slopes are negative reciprocals:
15
Example 3Find the slope-intercept form of the
equation of the line that passes through
the point (2, –1) and is parallel to the line
2x – 3y = 5.
16
Example 4Find the slope-intercept form of the
equation of the line that passes through
the point (2, –1) and is perependicular to
the line 2x – 3y = 5.
17
Extrapolation vs. InterpolationLinear Extrapolation
Use the equation of a line to estimate a point outside the two given points.
Linear InterpolationUse the equation of a line to estimate a point between the two given points.
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Additional Example
The number of gallons of gas left in your
gas tank can be approximated by a linear
function of the number of miles you have
driven. Suppose you have driven 123 mi
since your last fill-up and you have 13 gal
left in your tank. Your car uses 0.05
gal/mi (or 20 mi/gal).19
Additional Example (cont.)a. Write an equation in point-slope form
relating the number of miles driven since you filled the tank and the number of gallons left in the tank. Rewrite the equation to express gallons as a function of miles.
b. Use the answer found in part “a” to determine how many gallons your car’s tank holds.
c. You want to fill up before the amount left drops below 1 gallon. How much farther can you drive before you must fill up again?
20
Homework 1.1Worksheet 1.1
# 1, 19, 25, 29, 35, 43, 51,
55, 59, 65, 68, 71, 75 – 78
(matching), 83, 95, 96, 97
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