Functions and Graphs
2.1 Graphing and Intercepts
Reading the Coordinate Plane
Graphing gives us a visual way to see how inputs and outputs are connected.
Definition: Coordinate Pair
An ordered pair $(x,y)$ tells us to move $x$ units horizontally and $y$ units vertically from the origin.
Definition: Intercepts
For an equation in $x$ and $y$:
- The x-intercept(s) occur where $y=0$.
- The y-intercept occurs where $x=0$.
Example: Plot points and identify quadrants
Plot the points $(3,2)$, $(-4,1)$, and $(-2,-3)$.
Show Solution
- $(3,2)$ is in Quadrant I.
- $(-4,1)$ is in Quadrant II.
- $(-2,-3)$ is in Quadrant III.
The sign pattern tells the quadrant quickly:
- $(+,+)$: Quadrant I
- $(-,+)$: Quadrant II
- $(-,-)$: Quadrant III
- $(+,-)$: Quadrant IV
MyOpenMath: Practice: plotting points and quadrants (placeholder)
Finding Intercepts Algebraically
Intercepts are usually the first features we compute before sketching a graph.
Example: Find intercepts of a line
Find the intercepts of
$$2x+3y=12.$$
Show Solution
For the x-intercept, set $y=0$:
$$2x=12\Rightarrow x=6.$$
So the x-intercept is $(6,0)$.
For the y-intercept, set $x=0$:
$$3y=12\Rightarrow y=4.$$
So the y-intercept is $(0,4)$.
Example: Find intercepts of a quadratic
Find the intercepts of
$$y=x^2-5x+6.$$
Show Solution
For y-intercept, set $x=0$:
$$y=6,$$
so $(0,6)$.
For x-intercepts, set $y=0$:
$$x^2-5x+6=0=(x-2)(x-3).$$
So $x=2$ or $x=3$, giving $(2,0)$ and $(3,0)$.
