1. Review of Algebra

2. Equations & Inequalities

3. Functions & Graphs

4. Polynomial Functions

5. Rational Functions

6. Exponential and Logarithmic Functions

7. Systems of Equations and Inequalities

8. Matrices and Determinants

9. Conic Sections

10. 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)

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

Graph Functions Involving a Sequence of Transformations

Transformations of Functions

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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