Grade 12 Pre-Calculus Mathematics
[MPC40S]
Chapter 2
Radical Functions
Outcomes
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12P.R.13. Graph and analyze radical functions (limited to functions involving one radical)
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Chapter 2: RADICAL FUNCTIONS
2.1 β Radical Functions and Transformations Radical Function: ______________________________________________________ ______________________________________________________________________ ______________________________________________________________________ Example #1
Sketch the graph of the function π¦ = βπ₯. State the domain and range of the radical function.
π₯ π¦
Domain: __________________________ Range: ___________________________
General Transformation Model for Radical Functions You can graph a radical function of the form ______________________________
by transforming the graph of π¦ = βπ₯ based on the values of π, π, β, and π. The effects of changing parameters in radical functions are the same as the effects of changing parameters in other types of functions.
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Example #2
Sketch the graph of the following radical functions.
a) π¦ = βπ₯ + 2 b) π¦ = βπ₯ β 1 Example #3
Using the given equations, explain in words the transformations needed to transform
the graph of π¦ = βπ₯. Then sketch each graph and state the domain and range.
a) π¦ = β2βπ₯ + 1 ________________________________ ________________________________ ________________________________ Domain: ____________________ Range: _____________________
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b) π¦ = β2π₯ β 4 ________________________________ ________________________________ ________________________________ Domain: ____________________ Range: _____________________
c) π¦ + 1 = ββπ₯ + 3 ________________________________ ________________________________ ________________________________ Domain: ____________________ Range: _____________________
MPC40S Date: ____________________
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Example #4
Determine the equation of the following function written in the form
π¦ = πβπ(π₯ β β) + π
The point ____________ on the graph
of π¦ = βπ₯ is not affected by any stretches or any reflections. Use the image point of ____________ to find the value of _______ and _______ Answer: _________________________
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Example #5
Determine the equation of the following function written in the form
π¦ = πβπ(π₯ β β) + π
Answer: _________________________
MPC40S Date: ____________________
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Chapter 2: RADICAL FUNCTIONS
2.2 β Square Root of a Function Example #1
Sketch the following graphs on the same Cartesian plane.
π¦ = π₯ + 2 and π¦ = βπ₯ + 2
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π¦ = π₯ + 2 π¦ = βπ₯ + 2
x β intercept
y β intercept
Domain
Invariant Points
Steps to sketching π¦ = βπ(π₯) given the graph of π¦ = π(π₯) 1. Cross out the part of the graph where the y values are negative. 2. Place a point on the graph where π¦ = 0. 3. Place a point on the graph where π¦ = 1. 4. To find other points on the graph, keep the π₯ values the same and take the square root
of the π¦ values. 5. Connect all points in a smooth curve. Note: In between the invariant points, the square root graph should be above the original graph. Where the π¦ values are greater than 1, the square root graph should be below the original graph.
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Example #2
Sketch the following graphs on the same Cartesian plane. State the domain and any invariant points.
π¦ = βπ₯ + 4 and π¦ = ββπ₯ + 4
Example #3
Sketch the following functions. State the domain and any invariant points.
a) π¦ = π₯2 β 1 and π¦ = βπ₯2 β 1
Domain
Domain
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b) π¦ = 4 β π₯2 and y= β4 β π₯2
c) π(π₯) = (π₯ β 1)2 + 3 and π¦ = βπ(π₯)
Domain
Domain
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Example #4
Using the graph of π¦ = π(π₯), sketch the graphs of π¦ = βπ(π₯). a) b) c) d)
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Chapter 2: RADICAL FUNCTIONS
2.3 β Solving Radical Equations Example #1
a) Solve the following equation algebraically: 0 = βπ₯ + 4 β 3
b) Using the graph provided, determine the x β intercept of the graph of: π¦ = βπ₯ + 4 β 3 Note: The ____________, ____________, and ____________________ of a function all represent the same thing, the plane where the function crosses the x β axis. For this example, this also represents the solution to the equation.
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Example #2
a) Solve the following equation algebraically.
βπ₯ + 5 = π₯ β 1 b) Solve the same equation graphically. To do this, letβs separate the equation into two individual functions and graph each. The solution will be the x value of the point of intersection between the two graphs.