Transcript Unit 5 - e 1 B

UNIT 5
RELATIONS, FUNCTIONS, AND GRAPHING
FUNCTIONS
• A relation is a set of ordered pairs. For example:
{(3, 2), (4, 5), (6, 8), (7, 1)}
• The relation can represent a finite set of ordered
pairs or an infinite set.
Domain
x-coordinate
independent variable
Range
y-coordinate
dependent variable
• The domain of a relation is the set of all
x-coordinates from the ordered pairs in a relation.
• The range of a relation is the set of all
y-coordinates from the ordered pairs in a relation.
• A function is a special relation in which each
member of the domain is paired with only one
member of the range.
• No two ordered pairs have the same first element
EXAMPLES
• Determine whether each set of ordered pairs
represents a function
• {(3, 2), (4, 5), (6, 8), (7, 1)}
• {(0, 5), (4, 3), (6, 5), (-7, -4)}
• {(1, 3), (4, 2), (4, 1), (5, 6)}
• {(6, -2), (11, -3), (14, 9), (-14, 11), (-14, 20), (21, -21)}
•
x
5
3
2
1
0
-4
-6
y
1
3
1
3
-2
2
-2
• Vertical line test – if a vertical line on a graph passes
through more than 1 point it is not a function
• A solution to an equation or inequality in two
variables is an ordered pair (x, y) that makes the
equation or inequality true.
EXAMPLES
–3x + 6y = 12
x + 5y ≥ 11
(-4, 0)
(2, 1)
3y – 5x = 4
5y < 3x
(-2, -2)
(-1, -3)
GRAPHING LINEAR FUNCTIONS
Linear Equations
•
•
•
•
Equations whose graphs are a straight line
Find five values for the domain and make a table
Plot each ordered pair
Draw a straight line through the points
Label the line with the original equation
EXAMPLE 1
x
2x + 1
y
y
(x, y)
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1
2
3
4
5
x
EXAMPLE 2
x
x-4
y
y
(x, y)
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1
2
3
4
5
x
Standard Form of a Linear
Equation
where A, B, and C are real
numbers, and A and B
cannot both be zero.
EXAMPLE 3
• Graph 2x + y = 5
y
5
4
3
x
y
(x, y)
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1
2
3
4
5
x
WARM-UP
• Is it a function?
• Is it a function?
• {(6, -2), (11, -3), (14, 9), (-14, 11), (-14, 20), (21, -21)}
• Express 𝑥 −3 ≤ 𝑥 < 3 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 in roster
form.
HORIZONTAL & VERTICAL LINES
y=c
A _________________ line
parallel to the _____axis.
The y-coordinate has
the same value.
x=c
A _______________line
parallel to the____-axis.
The x-coordinate has
the same value.
EXAMPLES
y
y
5
5
4
4
3
3
2
2
1
1
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
–5 –4 –3 –2 –1
–1
–2
–2
–3
–3
–4
–4
–5
–5
1
2
3
4
5
x
The cost of renting a car for a day is $64.00 plus
$0.25 per mile. Let x represent the number of miles
the car was driven and let y represent the rental
cost, in dollars, for a day.
a. Write an equation for the rental cost of the car in
terms of the number of miles driven.
b. Graph the equation
EXAMPLES
• What must be the value of k be if (k, 4) lies on the
line 3x + y = 10?
• What must the value of k be if (5, -3) lies on the line
y–x=k
• Find k such that (k, 5) is a solution of 3x + 2y = 22
SLOPE OF A LINE
• Slope measures the steepness of a line by
comparing the rise to the run.
• The rise is the change in the “y” values while the run
is the change in the “x” values.
• Slope is often called the rate of change.
• We represent slope with the letter m.
Positive Slope
Negative Slope
Zero Slope
Undefined
SLOPE FORMULA
𝑆𝑙𝑜𝑝𝑒 𝐹𝑜𝑟𝑚𝑢𝑙𝑎:
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
EXAMPLES
• Find the slope of the line that is determined by the
points (-2,4) and (4,2).
• Find the slope of the line that goes through the
points (3, -3) and (2, -3).
EXAMPLES
• Draw the line that goes through the point (-2,1) and
3
has a slope of .
2
Graph the line that has a slope of -3 and goes through the point (1,3).
WARM-UP
• Find k such that (k, 5) is a solution of 3x + 2y = 22
• Find the slope of the line that goes through the
points (2, -4) and (-3, -3)
PARALLEL & PERPENDICULAR LINES
• Parallel lines never intersect, therefore, the slopes of
parallel lines are the same.
• Perpendicular lines intersect to form right angles.
The slopes of perpendicular lines are negative
reciprocals.
• When you multiply negative reciprocals, the
product is -1.
• When writing a negative reciprocal just think “flip
and change the sign”.
GUIDED PRACTICE
• 1) 𝑦 = 2𝑥 + 3
•
•
•
• 3) 2𝑥 − 3𝑦 = 6
•
•
1
2
2) − 𝑥 − 2 = 𝑦
4) 𝑦 = −4
WARM - UP
• Find the slope of the line that goes through the points
(2, -4) and (-3, -3)
1
• Are the lines 𝑦 = 2𝑥 + 3 and − 𝑥 − 2 = 𝑦 perpendicular?
2
Justify your answer.
• What can you tell me about the slopes of two lines that
are parallel?
GRAPHING LINEAR EQUATIONS USING
INTERCEPTS
• The x intercept is the point at which a function
crosses the x-axis.
• The y intercept is the point at which a function
crosses the y-axis.
• If we know these two points, we can graph a line.
KEY POINTS
• Y-intercept: x value is 0; (0,y)
• X-intercept: y value is 0; (x,0)
• To find the x-intercept, substitute 0 in for y and
evaluate
• To find the y-intercept, substitute 0 in for x and
evaluate
EXAMPLES
WARM-UP
• If line A has a slope of 2 and line B is parallel to it,
what is the slope of line B?
• What are the x and y intercepts of the line with the
equation y = 4x – 2?
• What does the graph of y = - 5 look like?
SLOPE INTERCEPT FORM
• 𝑦 = 𝑚𝑥 + 𝑏
• We call the above equation slope intercept form
because the m represents the slope and the b is the
y-intercept.
• If we have the slope and the y-intercept, we can
graph the line.
EXAMPLES
WRITING EQUATIONS IN SLOPE
INTERCEPT FORM
• Given the slope and a point, we can write an
equation in slope intercept form and then graph
the line.
• Method 1:
• Substitute the x and y coordinate into y = mx + b
• Evaluate to solve for “b”
• “Put it all together” in slope intercept form
EXAMPLE 1
• Slope = ½ and goes through the point (2,-3)
EXAMPLE 2
• Write the equation of the line with slope of 2 that
goes through the point (4,6)
EXAMPLE 3
• Write the equation of the line that is parallel to
y = 3x-1 and goes through the point (0,4)
EXAMPLE 4
• Write the equation of the line that is perpendicular
to 7x – 2y = 3 and goes through the point (4, -1)
POINT SLOPE FORM
• Method 2:
• Point Slope Form: 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
• Substitute the x and y coordinates and slope into
the equation
• Evaluate to get slope intercept form (y = mx + b)
EXAMPLE 5
• Write the equation of the line that has a slope of 4
and goes through the point (3, 5) then graph the
line.
• If we are given two points, we can still write the
equation:
1. Find the slope using the points given
2. Substitute the coordinates and slope
3. Evaluate
4. Equation should now be in slope intercept form
EXAMPLE 6
• Write the equation of the line that goes through the
points (-3, -4) and (-2, -8).
EXAMPLE 7
• Write the equation of the line that goes through the
points (2, 0) and (0, -1).