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Graphs of Equations
MATH 109 - PrecalculusS. 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
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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
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
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
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
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
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
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
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
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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