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

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

Sketching Equations

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

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

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

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

Finding x and y-intercepts of Equations

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

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

Using Symmetry to Sketch Equations

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

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|

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

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

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

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)

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

Finding Equations of Circles & Sketching Circles

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

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

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

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

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

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

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)