H. Algebra 2
10.3 Notes
10.3 Graphing Cube Root Functions
Date: ___________
Explore. Graphing and Analyzing the Parent Cube Root Function
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Parent Cube Root Function: 𝑓(𝑥) = √𝑥
A) Fill in the table and plot the points on the graph. Draw a smooth curve through the points to
sketch the graph.
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B) Give the domain and range of 𝑓(𝑥) = √𝑥.
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C) Does the graph of 𝑓(𝑥) = √𝑥 have any symmetry?
Graphing Cube Root Functions
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Learning Target G: I can graph cube root functions of the form 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘 or
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1
𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘 by hand.
Given each cube root function, describe the transformations applied to the parent graph and
find the point of symmetry and 2 points on each side to graph the function.
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A) 𝑔(𝑥) = 2 √𝑥 − 3 + 5
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3
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B) 𝑔(𝑥) = √2 (𝑥 − 10) + 4
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C) 𝑔(𝑥) = √𝑥 − 3 + 6
Writing Cube Root Functions
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Learning Target H: I can write a cube root function of the form 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘 or
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𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘 given its graph.
𝟑
𝟑
𝟏
Write a function of the form 𝒈(𝒙) = 𝒂√𝒙 − 𝒉 + 𝒌 or 𝒈(𝒙) = √𝒃 (𝒙 − 𝒉) + 𝒌 that matches each
graph.
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A) 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘
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B) 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘
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C) 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘
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D) 𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘
Modeling with Radical Functions
Learning Target I: I can model and evaluate a real-life situation using a radical function.
Square and Cube Root Functions can be used to model real-world situations. In addition, we can use these
models to investigate average rate of change.
Average Rate of Change of a function f(x) over an interval from 𝑥1 to 𝑥2 :
** this is the formula we use for _________________ of a line.
For A and B, use a calculator to evaluate the model at the indicated points, and connect the points with
a curve to complete the graph of the model. Calculate the average rates of changed of the first and last
intervals and explain what the rate of rate represents.
A) A car with good tires in on a dry road. The speed, in miles per hour, from which the car can stop in
a given distance d, in feet, is given by 𝑠(𝑑) = √96𝑑. Use distances of 20, 40, 60, 80, and 100 feet.
First, find and plot the points for the given x-values to draw the graph of the model.
Second, find the average rate of change on the intervals.
First:
Last:
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B) Water draining from a tank at an average speed, s, in feet per second, characterized by the
function 𝑠(𝑑) = 8√𝑑 − 2, where d is the depth of the water in the tank in feet. Evaluate the function
for depths of 2, 3, 4, and 5 feet.
d
s
2
3
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C) The velocity of a 1400-kilogram car at the end of a 400-meter run is modeled by the function
𝑣 = 15.2 3√𝑝, where v is the velocity in kilometers per hour and p is the power of its engine in
horsepower.
Graph the function and examine its average rate of change over the equal intervals of (0, 60),
(60, 120), and (120, 180). What is happening to the average rate of change as the p-values of the
intervals increase?
p
v
0
60
120
180
Use the function to find the velocity when p is 100 horsepower.
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