1. MATH 109 - Precalculus S. Rook
Graphs of Equations
MATH Precalculus
S. Rook

2. Overview Section 1.2 in the textbook: Sketching Equations
Finding x and y-intercepts of Equations
Using Symmetry to Sketch Equations
Finding Equations of Circles & Sketching Circles

3. Sketching Equations

4. Sketching Equations
Given an equation, we can pick values for one of the variables and solve for the other
E.g. Given y = -x2  when x = -2, y = -4
Thus, (-2, -4) lies on the graph of y = -x2
By repeating the process a few times, we obtain a graph of the equation
Usually 3 to 4 points are satisfactory
Pick both positive and negative values

5. Equations & Shapes of Graphs
We can often determine the shape of a graph based on its equation
Important to acquire this skill
Equations of the form:
y = mx + b are lines y = ax2 + bx + c are U-shaped

6. Equations & Shapes of Graphs (Continued)
y = |mx + b| are v-shaped

7. Sketching Equations
Ex 1: Sketch the equation: a) y = 2x – 1 b) y = -x2 + x c) y = 1 + |x – 3|

8. Finding x and y-intercepts of Equations

9. Finding Intercepts of Equations
x-intercept: where the graph of an equation crosses the x-axis
Written in coordinate form as (x, 0)
To find, set y = 0, and solve for x:
May entail solving a linear or quadratic equation
y-intercept: where the graph of an equation crosses the y-axis
Written in coordinate form as (0, y)
To find, set x = 0, and solve for y

10. Finding Intercepts of Equations (Example)
Ex 2: For each equation, find the a) y-intercept(s) b) x-intercept(s): a) b) c) d)

11. Using Symmetry to Sketch Equations

12. Symmetry
Knowing that an equation has symmetry means that we can use reflections to help us graph it
Symmetry is also helpful when asked to predict the behavior or shape of an equation
Most common types of symmetry:
Symmetry about the y-axis
Symmetry about the x-axis
Symmetry about the origin

13. Symmetry about the y-axis
Given an equation containing the point (x, y), the equation is symmetrical about the y-axis IF it also contains the point (-x, y)
Substituting -x for x into the equation does NOT change it
Ex: y = |x|

14. Symmetry about the x-axis
Given an equation containing the point (x, y), the equation is symmetrical about the x-axis IF it also contains the point (x, -y)
Substituting -y for y into the equation does NOT change it
Ex: x = -y2

15. Symmetry about the Origin
Given an equation containing the point (x, y), the equation is symmetrical about the origin IF it also contains the point (-x, -y)
Substituting -x for x & -y for y into the equation does NOT change it
Reflects over the line y = x
Ex: y = x3

16. Determining Symmetry (Example)
Ex 3: Use algebraic tests to check for symmetry with respect to both axes and the origin: a) x – y2 = 0 b) xy = 4 c) y = x4 – x2 + 3 d) y = 5x – 1

17. Using Symmetry to Sketch a Graph
If an equation is symmetric to the y-axis:
Get points using either x ≥ 0 or x ≤ 0
Obtain additional points by taking the opposite of x and keeping y the same (-x, y)
If an equation is symmetric to the x-axis:
Get points using either y ≥ 0 or y ≤ 0
Obtain additional points by taking the opposite of y and keeping x the same (x, -y)
If an equation is symmetric to the origin:
Get points using either x >= 0 or x <= 0
Obtain additional points by taking the opposite of both x and y (-x, -y)

18. Using Symmetry to Sketch a Graph (Example)
Ex 4: Use symmetry to sketch x = y2 – 5

19. Finding Equations of Circles & Sketching Circles

20. Standard Equation of a Circle
Circle: the set of all points r units away, where r is the radius, from a point (h, k) called the center
Given the radius and the center, we can construct the standard equation of a circle:
where:
(h, k) is the center
r is the radius

21. Sketching a Circle To sketch a circle: Plot the center (h, k)
From (h, k), plot four more points:
r units up
r units right
r units down
r units left
Complete the sketch

22. Standard Equation of a Circle (Example)
Ex 5: Write the standard form of the equation of the circle with the given characteristics: a) Center (2, -1); radius 4 b) Center (0, 0); radius 4

23. Standard Equation of a Circle (Example)
Ex 6: Find the center and radius of the circle, and sketch its graph: a) b)

24. General Equation of a Circle
An equation in the form x2 + y2 + Ax + By + C = 0 (A, B, and C are constants) is known as the general equation of a circle
Notice that the right side of the general equation is set to 0
To extract the center and radius:
Complete the square on x and then on y to convert the general equation to the standard equation
We will review the process of completing the square in the next example

25. Standard Equation of a Circle (Example)
Ex 7: Find the center and radius of the circle: a) b)

26. Summary
After studying these slides, you should be able to:
Sketch a graph, determining the shape from its equation if possible
Find x and y-intercepts of an equation
Determine symmetry of an equation
Find and sketch equations of circles
Additional Practice:
See the list of suggested problems for 1.2
Next lesson
Linear Equations in Two Variables (Section 1.3)