1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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In Exercises 17-32, use the graph of y = f(x) to graph each function g.  g(x) = 2f(x+2) − 1

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x)=2f(x-1)

Format for Reflection of Function Graphs Across the x-Axis

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = -½ ƒ ( x + 2) —2

Graphing a Cotangent Function, EX 1

In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. r(x) = 2√(x + 2)<img src="https://lightcat-files.s3.amazonaws.com/problem_images/2ac314f6c846bdb7-1673615104543.jpg" alt="" />

Graphing a Rational Function - Example 1

Format for Reflection of Function Graphs Across the y-Axis

In Exercises 55–59, use the graph of to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/d6497834d7f11c1c-1673614902277.jpg" alt="" /> g(x) = -f(2x)

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/83f8b85fc7f03b5e-1678788096081.jpg" alt="" />

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -(x − 2)²

Horizontal Stretch or Compression of the Graph of a Function

Vertical Stretch or Compression of the Graph of a Function

Graphing the Greatest Integer or Floor Function

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛x+2

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2625b926e32e8c1d-1676842896486.jpg" alt="" /> g(x) = ½ f(x)

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/e5aefc5fdf3a584b-1675215542178.jpg" alt="" /> g(x) = −ƒ( x/2) +1

In Exercises 17-32, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/33d8326ea1a04f2b-1676919502216.jpg" alt="" /> g(x) = f(x-1)

In Exercises 17-32, use the graph of y = f(x) to graph each function g.  g(x) = f(x + 1) − 2

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

Understand horizontal scaling of quadratic equations

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)^3

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/1be7e25606340ef4-1681141549114.jpg" alt="" /> 
(a) y = ƒ(x) +3

Graphing a Cotangent Function, EX 2

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. r(x) = (x − 2)³ +1

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x) = -f(x + 1) − 1

Graphing Trigonometric Functions - Graph Transformations P1

Graphing a Sine Function  EX 2

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2

Graph absolute value functions

Graphing a Cosecant Function , EX 1

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=2√x+1

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -|x + 4| +2

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7ea891059d79474b-1675171880601.jpg" alt="" /> g(x) = f(x)+1

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. ∛(-x+2)

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x) = √(x+1)-1

Application of quadratic function transformations

In Exercises 17-32, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2f9db6cf2f229fd4-1675162878495.jpg" alt="" /> g(x) = f(x) - 1

Consider the following nonlinear system. Work Exercises 75 –80 in order.
y = | x - 1 |
y = x^2 - 4

How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = |x + 3| - 2

Review of the Graphs of Some Common Functions

Graph cube root functions

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/5c03b3b298a3caaf-1675215395491.jpg" alt="" /> g(x) = f(x/2)

In Exercises 17-32, use the graph of y = f(x) to graph each function g.  g(x) = f(-x)

Understand vertical scaling of quadratic equations

Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4. y=x^2+5

Format for the Vertical Stretch or Compression of the Graphs of Functions

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = -|x+3|

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = f(x)+2

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = -x³

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = (1/2)(x − 2)³ – 1

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/b357f7d30c70b131-1681143782839.jpg" alt="" /> 
(c) y = ƒ(x+3) - 2

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = (1/2)f(2x)

Format for Vertical Shifts of Graphs of Functions

Graph each function. See Examples 1 and 2. g(x)=(1/2)x^2

Graphing a Tangent Function - EX 1

Graph each function. See Examples 1 and 2. h(x)=√(4x)

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/9dfbb13a3fd17534-1678786059088.jpg" alt="" />

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = (1/2) (x − 1)² – 1

Graphing Exponential Functions w/ Graph Transformations

Graphing  y = 30sin(pi/10)t  + 35

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2645054ce456534c-1676842056658.jpg" alt="" /> g(x) = 2f(x)

In Exercises 17-32, use the graph of y = f(x) to graph each function g.  g(x) = f(x+1)

Apply vertical scaling transformations to linear equations

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. ∛(-x-2)

Graph Functions Involving a Sequence of Transformations

Transformations of Functions

Trigonometric Functions and Graphing: Amplitude, Period, Vertical and Horizontal Shifts, Ex 2

Graph each function. See Examples 1 and 2. ƒ(x)=3|x|

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/74c8600b8a68f935-1674737656688.jpg" alt="" /> g(x) = f(x+1)

Transforming Algebraic Functions: Shifting, Stretching, and Reflecting

Perform Transformations on a Given Graph

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x-2)

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/85db4f2ed9dd408f-1676902533777.jpg" alt="" /> g(x) = -f(x) +3

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x) = -f(x-1) + 1

Graph of y = e ^ (x + 3) using Graph Transformations

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. r(x) = -(x + 1)^2

Horizontal Shift of a Graph

Describe how the graph of each function can be obtained from the graph of ƒ(x) = |x|. 
g(x) = -|x|

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -2(x+2)²+1

Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x^2

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)

Graph each function. See Examples 1 and 2. ƒ(x)=-√-x

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x).
across the y-axis

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7e506169afe94e10-1681142751820.jpg" alt="" /> 
(b) y = ƒ(x-2)

Graphing f(x) = (1/x) + 5 ; Rational Functions and Graph Transformation

In Exercises 17-32, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ea01b68a3062d535-1675120346473.jpg" alt="" /> g(x) = ½ f(x)

In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x + 3)

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = f(x+2)

Graphing Exponential Functions

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x|+3

Graph each function. See Examples 1 and 2. ƒ(x)=-3|x|

Horizontal And Vertical Graph Stretches and Compressions Part 2 of 3

Graphing a Cosecant Function , EX 2

Write an Equation From a Description of Transformations

Graphing Sine and Cosine with Phase (Horizontal) Shifts, Example 2

Combined Shifting of a Function

Graph each function. See Examples 1 and 2. h(x)=|-(1/2)x|

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x) =(1/2) f(2x)

In Exercises 55–59, use the graph of to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/9f626f58538c55c4-1673620274795.jpg" alt="" /> g(x) = f(x + 2) + 3

In Exercises 17-32, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2f945dd6fe72c8bc-1675169715772.jpg" alt="" /> g(x) = f(-x)+1

Graph each function. See Examples 1 and 2. ƒ(x)=2/3|x|

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/374122806c8cd167-1675180272586.jpg" alt="" /> g(x) = -f(2x) - 1

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=-√(x + 1)

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x+3|

Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4. x^2+y^2=12

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = x³/2

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=√(-x+1)

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/de1a86c75f748fef-1676903959113.jpg" alt="" /> g(x) = f(-x)

Graphs to Know for Your Algebra Class

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/174e28204532a92b-1676902471648.jpg" alt="" /> g(x) = f(-x)+3

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/e3b8edaa2a9139a8-1679508146359.jpg" alt="" /> 
(d) y = |ƒ(x)|

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛(x-2)

Finding Amplitude, Period, Horizontal and Vertical Shifts of a Trig Function EX 1

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x+1)

Graphing Using Graph Transformations - Example 2.

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x^2+2

The graph of y=|x-2| is symmetric with respect to a vertical line. What is the equation of that line?

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. -∛(x+2)

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)-1

In Exercises 45-52, use the graph of y = f(x) to graph each function g.  g(x) = f(x-1) – 1

In Exercises 17-32, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/965f3e1da68b6fb9-1676907720832.jpg" alt="" /> g(x) = f(x-1)+2

Graphing a Sine Function  EX 4

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (5, -3)

Reflect linear equations

In Exercises 67–69, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. r(x) = (1/2) |x + 2|<img src="https://lightcat-files.s3.amazonaws.com/problem_images/fdfada08352f5f0d-1673614475756.jpg" alt="" />

Graphing a Cosine Function  EX 4

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = -f(x+2)

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (x − 2)²

Reflect points over the x and y axes

Graphing a Parabola

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x+2) - 2

Consider the following nonlinear system. Work Exercises 75 –80 in order.
y = | x - 1 |
y = x^2 - 4

How is the graph of y = x^2 - 4 obtained by transforming the graph of y = x^2?

Graph square root functions using transformations

Graphing a Tangent Function - EX 2

In Exercises 17-32, use the graph of y = f(x) to graph each function g.  g(x) = -f(x)+1

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=-3(x-2)^2+1

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²

Reflection Across the x-Axis

Graphing a Cosine Function  EX 3

Vertically translate quadratic equations

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (-4, -2)

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. g(x)=(x+2)^2

Vertical Shift of a Graph

In Exercises 33-44, use the graph of y = f(x) to graph each function g.  g(x) = -(1/2)f(x+2)

Horizontal and Vertical Graph Transformations

Graphing a Secant Function, EX 1

Graphing Quadratic Functions - Example 1

M8: Reflection Across the y-Axis

Format for Horizontal Shifts of Graphs of Functions

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|x+3|

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x).
across the x-axis

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/8af8dcb15d02aa4b-1681149311064.jpg" alt="" />

In Exercises 1-16, use the graph of y = f(x) to graph each function g. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/47e915ef4e9744a6-1676841378623.jpg" alt="" /> g(x) = f(x-1) - 2

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = (x − 3)^3

Applying Combinations of Transformations

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