Download Lecture notes for Section 2.1

Int. Alg. Notes
Section 2.1
Page 1 of 7
Section 2.1: Rectangular Coordinates and Graphs of Equations
Big Idea: The values produced by an algebraic expression can be represented visually by graphing the answers
produced vs. the numbers put into the calculation.
Big Skill: You should be able to graph the output of linear and nonlinear algebraic equations using the pointplotting method, and identify intercepts of a graph.
Quick verbal review of the rectangular coordinate system and plotting points…
x-axis
y-axis
rectangular/Cartesian coordinate system
xy-plane
coordinate axes
ordered pair (x, y)
coordinates of a point P
quadrants
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 2 of 7
Definition: Equation in Two Variables
An equation in two variables (like x and y) is a statement in which the algebraic expressions involving x and y
are equal. The expressions are called the sides of the equation.
Definition: Graph of an Equation in Two Variables
The graph of an equation in two variables (like x and y) is the set of all ordered pairs (x, y) in the xy-plane
that satisfy the equation.
When we want to visually summarize all the possible answers to an equation, we graph the equation. The
easiest way to do this is to calculate a few points by hand, and then plot those points on a graph. Usually, it is
easy enough to “connect the dots” to fill in the rest of the graph.
One thing to remember is that it is easiest to make a table of values after you solve the equation for one of the
variables.
Also, a trick for graphing lines is to compute two points only, then connect the points with a straight line.
Practice:
1. Write an algebraic equation for how much you make if you work for h hours at a rate of $15 per hour.
Then graph this equation.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 3 of 7
2. A cell phone package has a flat fee of $10 a month, plus 10 cents a minute for every minute you talk.
Write an equation for the cost of this package, and then graph it.
3. Graph the equation y  3x  1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 4 of 7
4. Graph the equation 2 x  3 y  1
5. Graph the equation y  x 2  3
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 5 of 7
6. Graph the equation x  y 2
7. Graph the equation y  x3  x
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 6 of 7
One important feature of graphs is where they cross the coordinate axes.
An x-intercept of a graph is the x-coordinate of a point on the graph that crosses or touches the x-axis.
A y-intercept of a graph is the y-coordinate of a point on the graph that crosses or touches the y-axis.
y




      







x


y = x^4 - 5x^2 + 4

This graph has x-intercepts at x = -2, x = -1, x = +1, and x = +2, and a y-intercept at y = 4.
Practice:
8. Identify the intercepts of the graph below.
y










x




Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 2.1
Page 7 of 7
One advantage of graphing an equation is that you can extract meaning from the graph that is hard to extract
from the equation. A lot of this meaning comes from looking at the intercepts and at extreme points of the
graph, and then relating those points to the context of the problem.
Practice:
9. The vertical height of a ball y (as measured in meters) thrown straight up into the air at 10 meters per
second from a height of 6 meters is predicted as a function of time t (as measured in seconds) by the
1
equation y    9.8  t 2  10t  6 . The graph is shown below.
2
a. What is the height of the ball after 0.6 second?
b. When does the ball reach that same height again?
c. Interpret the meaning of the intercepts.
d. What is the highest height reached by the ball? When does it get to this height?
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.