7.1 R eview of Graphs and Slopes of Lines Standard form of a linear equation: The graph of any...
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Transcript of 7.1 R eview of Graphs and Slopes of Lines Standard form of a linear equation: The graph of any...
7.1 Review of Graphs and Slopes of Lines
• Standard form of a linear equation:
• The graph of any linear equation in two variables is a straight line. Note: Two points determine a line.
• Graphing a linear equation:1. Plot 3 or more points (the third point is used
as a check of your calculation)2. Connect the points with a straight line.
CByAx
7.1 Review of Graphs and Slopes of Lines
• Finding the x-intercept (where the line crosses the x-axis): let y=0 and solve for x
• Finding the y-intercept (where the line crosses the y-axis): let x=0 and solve for y
Note: the intercepts may be used to graph the line.
7.1 Review of Graphs and Slopes of Lines
• If y = k, then the graph is a horizontal line (slope = 0):
• If x = k, then the graph is a vertical line (slope = undefined):
7.1 Review of Graphs and Slopes of Lines
• Slope of a line through points (x1, y1) and (x2, y2) is:
• Positive slope – rises from left to right.Negative slope – falls from left to right
run
rise
xx
yym
)(
)(
in x change
yin change
12
12
7.1 Review of Graphs and Slopes of Lines
• Using the slope and a point to graph lines:Graph the line with slope passing through the point (0, 0)
Go over 5 (run) and up 3 (rise) to get point (5, 3) and draw a line through both points.
run
risem
5
3
5
3
7.1 Review of Graphs and Slopes of Lines
• Finding the slope of a line from its equation:
1. Solve the equation for y2. The slope is given by the coefficient of x
• Parallel and perpendicular lines:1. Parallel lines have the same slope2. Perpendicular lines have slopes that are
negative reciprocals of each other
7.1 Review of Graphs and Slopes of Lines
• Example: Decide whether the lines are parallel, perpendicular, or neither:
1. solving for yin first equation:
2. solving for yin second equation:
3. The slopes are negative reciprocals of each other so the lines are perpendicular
32
72
yx
yx
2
3232
m
xyyx
21
27
21
7272
mxy
xyyx
7.2 Review of Equations of Lines
• Standard form:
• Slope-intercept form:(where m = slope and b = y-intercept)
• Point-slope form: The line with slope m going through point (x1, y1) has the equation:
CByAx bmxy
)( 11 xxmyy
7.2 Review of Equations of Lines
• Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x + 3y = 6
1. solve for y to get slope of line
2. take the negative reciprocal to get the slope32
32 2
623632
mxy
xyyx
23m
7.2 Review of Equations of Lines
• Example (continued):
3. Use the point-slope form with this slope and the point (-4,5)
4. Add 5 to both sides to get in slope intercept form:
11
645
)4(5
23
23
23
23
xy
xxy
xy
23m
7.3 Functions Relations
• Relation: Set of ordered pairs:
Example: R = {(1, 2), (3, 4), (5, 1)}
• Domain: Set of all possible x-values
• Range: Set of all possible y-values
• What is the domain of the relation R?
7.3 FunctionsRelations
Domain:x-values(input)
Range:y-values(output)
Example: Demand for a product depends on its price.Question: If a price could produce more than one demand would the relation be useful?
7.3 Functions - Determining Whether a Relation or Graph is a Function
• A relation is a function if: for each x-value there is exactly one y-value– Function: {(1, 1), (3, 9), (5, 25)}– Not a function: {(1, 1), (1, 2), (1, 3)}
• Vertical Line Test – if any vertical line intersects the graph in more than one point, then the graph does not represent a function
7.3 Functions
• Function notation: y = f(x) – read “y equals f of x”note: this is not “f times x”
• Linear function: f(x) = mx + b
Example: f(x) = 5x + 3
• What is f(2)?
7.3 Functions - Graph of a Function
• Graph of
• Does this pass the vertical line test?What is the domain and the range?
xxf )(
7.3 Functions - Graph of a Parabola
Vertex
2)( xxf
7.4 Variation
• Types of variation:1. y varies directly as x:2. y varies directly as the
nth power of x:
3. y varies inversely as x:
4. y varies inversely as the nth power of x: nx
ky
nkxy
x
ky
kxy
7.4 Variation
• Solving a variation problem:1. Write the variation equation.
2. Substitute the initial values and solve for k.
3. Rewrite the variation equation with the value of k from step 2.
4. Solve the problem using this equation.
7.4 Variation• Example: If t varies inversely as s and
t = 3 when s = 5, find s when t = 5
1. Give the equation:
2. Solve for k:
3. Plug in k = 15:
4. When t = 5: 315515
5 sss
s
kt
155
3 kk
st
15
9.2 Review – Things to Remember
• Multiplying/dividing by a negative number reverses the sign of the inequality
• The inequality y > x is the same as x < y• Interval Notation:
– Use a square bracket “[“ when the endpoint is included
– Use a round parenthesis “(“ when the endpoint is not included
– Use round parenthesis for infinity ()
9.2 Review - Compound Inequalities and Interval Notation
Solve eachinequality for x:
Take the intersection:(why does the order change?)Express in interval notation:
422 and 1013 xx22 and 93 xx
1 and 3 xx
3,1
1 3
31 x
9.2 Review - Compound Inequalities and Interval Notation
Solve eachinequalityfor x:
Take the union:
Express ininterval notation
33-or 32 xxx1or 03 xx1or 3 xx
-1
1x
),1[
9.2 Review - Absolute Value Equations
• Solving equations of the form: kbax
35or 1
53or 33
143or 143
143
xx
xx
xx
x
9.2 Absolute Value Inequalities
• To solve where k > 0, solve the compound inequality (intersection):
• To solve where k > 0, solve the compound inequality (union):
Why can’t you say ?
kbax
kbaxk
kbax
kbaxkbax or
kbaxk
9.2 A Picture of What is Happening
• Graphs of
and f(x) = k
The part below the line f(x) = k is where
The part above the line f(x) = k is where
baxxf )(
)0,( ab
kbax
x
y
f(x) = k
kbax
9.2 Absolute Value Inequalities - Form 1
• Solving equations of the form:
1. Setup the compoundinequality
2. Subtract 4 all the wayacross
3. Divide by 3
4. Put into intervalnotation
kbax
135 x
1,35
143 x1431 x
335 x
9.2 Absolute Value Inequalities - Form 2
• Solving equations of the form:
1. Setup the compoundinequality
2. Subtract 4 all the wayacross
3. Divide by 34. Put into interval notation
What part of the real line is missing?
kbax
35or x 1 x
,1(), 35
143 x143or 143 xx
53or 33 xx
9.2 Absolute Value Inequalitythat involves rewriting
• Example:
Add 3 to both sides (why?):
Set up compound equation:
Add 2 all the way across:
Put into interval notation
132 x
22 x
222 x
40 x
4,0
9.2 Absolute Value Inequalities
• Special case 1 when k < 0:
Since absolute value expressions can never be negative, there is no solution to this inequality. In set notation:
435 x
9.2 Absolute Value Inequalities
• Special case 2 when k = 0:Since absolute value expressions can never be negative, there is one solution for this:
In set notation: What if the inequality were “<“?
035 x
53
35
035
035
x
x
x
x
53
9.2 Absolute Value Inequalities
• Special case 3:
Since absolute value expressions are always greater than or equal to zero, the solution set is all real numbers. In interval notation:
135 x
,
9.2 A Picture of What HappensWhen k is Negative
• Graphs of
and f(x) = k
never gets below the line f(x) = k so there is no solution toand the solution to is all real numbers
baxxf )(
)0,( ab
baxxf )(
x
y
f(x) = k
kbax kbax
9.2 Relative Error
• Absolute value is used to find the relative error of a measurement. If xt represents the expected value of a measurement and x represents the actual measurement, then
relative error in t
t
x
xxx
9.2 Example of Relative Error
• A machine filling quart milk cartons is set for a relative error no greater than .05. In this example, xt = 32 oz. so:
Solving this inequality for x gives a range of values for carton size within the relative error specification.
05.32
32
x
9.2 Solution to the Example
1. Simplify:
2. Change into acompound inequality
3. Subtract 1
4. Multiply by –32
5. Reverse the inequality
6. Put into interval notation
05.32
132
32 xx
05.32
105. x
95.32
05.1 x
4.306.33 x6.334.30 x
6.33,4.30