Transcript B-2 Linear Equationsx

Linear Relations and
Functions
B-2
Linear Equations
ACT WARM-UP
• Which of the following is equivalent to
(b)(b)(b)(b)(b) + (b)(b)(b)(b)(b)(b)?
• A) 11b
B) 11b
C) b + 11
D) b5 + b6
E) b11
• When multiplying like bases, keep the
base and add exponents. The answer is
D) b5 + b6.
Objectives
• Identify linear equations and functions.
• Write linear equations in standard form
and graph them.
Essential Question
How do you graph linear equations
written in standard form?
An equation such as x + y = 4 is called
a linear equation. A linear equation has
no operations other than addition,
subtraction, and multiplication of a
variable by a constant. The variables
may not be multiplied together or
appear in a denominator. A linear
equation does not contain variables with
exponents other than 1. The variables
are not under a radical sign and cannot
be an exponent.
Linear Equation
•
•
•
•
5x – 3y = 7
x = 9 (not a function)
6a = – 3b – 15
y = 0.5x
Not Linear
• 7m + 4n² = – 8
• y = √x + 5
• x + xy = 1
• y=
1
𝑥
A linear function is a function whose
ordered pairs satisfy a linear equation. Any
linear function can be written in the form
f(x) = mx + b, (commonly called slopeintercept form) where x is the independent
variable and m (slope) and b (y-intercept)
are constants. The graph of a linear
function is a straight line made up of the set
of all points that satisfy y = f(x).
State whether
Explain.
is a linear function.
Answer: This is a linear function because it is in the form
State whether
Explain.
is a linear function.
Answer: This is not a linear function because x has an
exponent other than 1.
State whether
Explain.
is a linear function.
Answer: This is a linear function because it can be written
as
State whether each function is a linear function.
Explain.
a.
Answer: yes;
b.
Answer: No; x has an exponent
other than 1.
c.
Answer: No; two variables are
multiplied together.
Meteorology The linear function
can be used to find the number of degrees Fahrenheit,
f (C), that are equivalent to a given number of degrees
Celsius, C.
On the Celsius scale, normal body temperature is
37C. What is normal body temperature in degrees
Fahrenheit?
Original function
Substitute.
Simplify.
Answer: Normal body temperature, in degrees
Fahrenheit, is 98.6F.
Meteorology The linear function
can be
used to find the distance m(s) in miles from a storm,
based on the number of seconds s that it takes to
hear thunder after seeing lightning.
a. If you hear thunder 10 seconds after seeing lightning,
how far away is the storm?
Answer: 2 miles
b. If the storm is 3 miles away, how long will it take to
hear thunder after seeing lightning?
Answer: 15 seconds
Standard Form of a Linear
Equation
• The standard form of a linear equation is
Ax + By = C, where
• A ≥ 0,
• A and B are not both zero at the same time,
• A, B, and C are integers, and
• A, B, and C do not have a GCF other than 1.
Write
in standard form. Identify A, B, and C.
Original equation
Add – 3x to each side.
Multiply each side by –1 so
that A  0.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Add – 2y to each side.
Multiply each side by –3 so that
the coefficients are all integers.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Add – 4 to each side.
Divide each side by 2 so that the
coefficients have a GCF of 1.
Answer:
and
Write each equation in standard form. Identify A, B,
and C.
a.
Answer:
and
b.
Answer:
and
c.
Answer:
and
You should be familiar with graphing a function by
making a table of values, seeing a pattern, and
connecting the points with a line or smooth curve.
Since two points determine a line, there are
quicker ways to graph a linear function. One way
is to find the points at which the graph intersects
each axis and connect them with a line. The ycoordinate of the point at which a graph crosses
the y-axis is called the y-intercept (0, b). The x
value at this point is zero. Likewise, the xcoordinate of the point at which it crosses the xaxis is the x-intercept (x, 0). The y value at this
point is zero.
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
The x-intercept is the value of x when
Original equation
Substitute 0 for y.
Add 4 to each side.
Divide each side by –2.
The x-intercept is –2. The graph crosses the
x-axis at (–2, 0).
Likewise, the y-intercept is the value of y when
Original equation
Substitute 0 for x.
Add 4 to each side.
The y-intercept is 4. The graph crosses the y-axis at (0, 4).
Use the ordered pairs to graph this equation.
Answer: The x-intercept is –2, and the y-intercept is 4.
(0, 4)
(–2, 0)
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
Answer: The x-intercept is –2, and the y-intercept is 6.
Essential Question
How do you graph linear equations
written in standard form?
Find the x- and y-intercepts by substituting 0
for x and y respectively. Plot the points on
the appropriate axis and connect with a
straight line extending beyond the points.
Math Humor
• Parent: Did you study your algebra lesson
at the family reunion?
• Student: Sure, it was a function with
relations.