STAT 0086 Ohio State University Graphing Linear Equations Algebra Questions

Homework 11 – Graphing Linear Equationswith Slope and Intercept
Name: _____________________________
1. Find the slope of the line that contains the points (−3, 5) and (−1, −4). Show your work.
1. _________
2. A teacher weighed 145 pounds in 1986 and weighs 190 pounds in 2007. What is the rate of
change in weight?
2. _________
3. A rocket is 1 mile above the earth in 30 seconds and 5 miles above the earth in 2.5 minutes.
What is the rocket’s rate of change in miles per second?
3. _________
4. Find the x and y-intercepts of the line. Show your work.
4. x-intercept: ( __ , 0)
−2𝑥 + 𝑦 = 8
y-intercept: ( 0, __ )
5. Find the slope and y-intercept of the line. Then, graph the line using that information.
2
𝑦 = 5𝑥 − 1
m = _______
y-intercept: (0, __ )
6. Find the slope and y-intercept of the line. Then, graph the line using that information.
−3𝑥 − 𝑦 = 5
m = ________
y-intercept: (0, __ )
7. Find the slope and y-intercept of the line. Then, graph the line using that information.
3𝑥 + 2𝑦 = 8
m = ________
y-intercept: (0, __ )
STAT0086 Co-Requisite – Lesson 10
Graphing Ordered Pairs and Linear Equations
How many solutions does each equation have?
a)
𝑥+1=8
b)
𝑥+𝑦 =3
An equation of the form 𝐴𝑥 + 𝐵𝑦 = 𝐶 is called a linear equation. A solution to a linear equation
is any pair of values that when plugged in for the variables makes the two sides equal.
Each solution can be written as an ordered pair, (𝑥, 𝑦). It is ordered because the 𝑥-value comes
first and the 𝑦-value comes second.
Ordered pairs can be graphed on a coordinate plane. The horizontal axis is the 𝑥-axis and the
vertical axis is the 𝑦-axis.
Example: Graph the ordered pairs on a coordinate plane.
(3, 4)
(−2, 1)
(−1, −3)
(1.5, −2)
(0, −4)
3
(− , 0)
2
A linear equation has infinitely many solutions. If all of those ordered pairs were graphed on a
coordinate plane, the result is a line. A linear equation can be graphed by finding any two
solutions, plotting the ordered pairs and drawing a line through the points.
Example: Graph the equations by finding two solutions.
a)
𝑥 − 2𝑦 = 8
b)
2𝑥 + 3𝑦 = 12
c)
−𝑥 + 3𝑦 = 9
d)
𝑦=8
e)
𝑥=4
STAT0086 Co-Requisite – Lesson 11
Graphing Linear Equations with Slope and Intercept
𝑥 – variable: Independent variable or Explanatory variable
𝑦 – variable: Dependent variable or Response variable
The graph of an equation tells us about the relationship between the independent and dependent
variables. This relationship is called the slope. It is also referred to as the average rate of
change. The steeper the slope (or graph of the line), the faster the dependent variable changes
given a certain change in the independent variable.
𝑆𝑙𝑜𝑝𝑒 = 𝑚 =
𝑟𝑖𝑠𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
=
𝑟𝑢𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
Slope Formula
Given any two points (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), the slope (or average rate of change) of the line that
contains the points is
𝑦 −𝑦
𝑚 = 𝑥2 − 𝑥1 .
2
1
Example: Find the slope of the line that contains the following points.
a)
(6, 0)
c)
(−4, 9)
(8, 10)
b)
(−3, 4)
(−1, −7)
(3, 9)
d)
(8, −1)
(8, 0)
Example: A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he
is at an altitude of 700 feet. What is the average rate of change?
Example: A scuba diver is 30 feet below the surface of the water 10 seconds after he entered the
water and 100 feet below the surface after 40 seconds. What is the scuba diver’s rate of change?
Intercepts
•
The 𝒙-intercept of the graph of a line is the point where the line crosses the 𝑥-axis. It
always has the form (𝑥, 0). To find the 𝑥-intercept, replace 𝑦 with 0 in the equation and
solve for 𝑥.
•
The 𝒚-intercept of the graph of a line is the point where the line crosses the 𝑦-axis. It
always has the form (0, 𝑦). To find the 𝑦-intercept, replace 𝑥 with 0 in the equation and
solve for 𝑦.
Example: Find the 𝑥 and 𝑦-intercepts for each equation.
a)
3𝑥 + 2𝑦 = 8
c)
𝑦 = 3𝑥 − 3
2
b)
−𝑥 + 2𝑦 = 5
d)
𝑦 = −4𝑥 + 9
Slope-Intercept Form:
𝑦 = 𝑚𝑥 + 𝑏
where 𝑚 is the slope of the line and (0, 𝑏) is the 𝑦-intercept of the line
Example: Identify the slope and intercept of the line. Then, graph the equation using the slope
and 𝑦-intercept.
a)
1
𝑦 = −3𝑥 + 4
b)
𝑦 = −𝑥 + 7
c)
2𝑥 + 9𝑦 = −9
d)
−4𝑥 + 5𝑦 = 12
STAT0086 Co-Requisite – Lesson 9
Solving and Graphing Inequalities
Solving an Inequality
The process of solving an inequality is very similar to the process of solving an equation.
However, when multiplying or dividing both sides by a negative value, you need to flip the
inequality symbol.
The solution to an inequality is different than the solution to an equation. For example, compare
the following solutions.
𝑥=3
𝑥≤3
Examples: Solve each inequality for the variable. Then, graph the solution set on a number line.
a)
4𝑥 + 6 ≥ 10
b)
−3𝑥 + 4 > 9
c)
𝑦 − 6 > 3𝑦 + 4
d)
5𝑦 + 8 ≤ 2𝑦 + 5
e)
−3(8 − 2𝑥) ≥ 2 + 3𝑥 − 10
f)
−4 < 5𝑥 + 1 < 16 g) 1 ≤ 3𝑥 − 5 ≤ 10