Graphs of Equations MATH 109 - Precalculus S. Rook.

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Graphs of Equations MATH 109 - Precalculus S. Rook

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

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

Graphs of Equations

MATH 109 - PrecalculusS. Rook

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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

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Sketching Equations

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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

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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

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Equations & Shapes of Graphs (Continued)

y = |mx + b| are v-shaped

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Sketching Equations

Ex 1: Sketch the equation:

a) y = 2x – 1b) y = -x2 + xc) y = 1 + |x – 3|

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Finding x and y-intercepts of Equations

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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• May entail solving a linear or quadratic equation

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Finding Intercepts of Equations (Example)

Ex 2: For each equation, find the a) y-intercept(s) b) x-intercept(s):

a) b)

c) d)

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23 42 xxy 12 xy

73 xy 12 xy

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Using Symmetry to Sketch Equations

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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

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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|

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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

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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

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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

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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)

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Using Symmetry to Sketch a Graph (Example)

Ex 4: Use symmetry to sketch x = y2 – 5

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Finding Equations of Circles & Sketching Circles

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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 centerr is the radius

222 rkyhx

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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

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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

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Standard Equation of a Circle (Example)

Ex 6: Find the center and radius of the circle, and sketch its graph:

a)

b)

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4

9

2

1

2

122

yx

11 22 yx

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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 example24

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Standard Equation of a Circle (Example)

Ex 7: Find the center and radius of the circle:

a)

b)

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0162822 yxyx

012422 xyx

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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)

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