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Precalculus Study Guide: Graphs and Transformations of Functions

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Q1. Sketch the graph of . Provide the domain, range, and evaluate and .

Background

Topic: Linear Functions and Transformations

This question tests your understanding of how to graph linear functions, identify their domain and range, and evaluate the function at specific values.

Key Terms and Formulas

  • Linear function: where is the slope and is the y-intercept.

  • Domain: All possible input values () for the function.

  • Range: All possible output values () for the function.

  • To evaluate , substitute into the function.

Step-by-Step Guidance

  1. Rewrite the function in standard linear form: can be simplified by combining like terms.

  2. Identify the slope () and y-intercept () from the simplified equation.

  3. Recall that the domain of any linear function is all real numbers, unless otherwise restricted.

  4. Similarly, the range of a non-horizontal linear function is also all real numbers.

  5. To evaluate and , substitute and into your simplified function, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

The simplified function is .

Domain:

Range:

Linear functions have no restrictions on domain or range, and evaluating at specific values is straightforward substitution.

Q2. Sketch the graph of . Identify the roots, extrema, intervals of increase and decrease.

Background

Topic: Quadratic Functions and Transformations

This question tests your ability to graph a transformed quadratic function, find its roots (x-intercepts), vertex (extremum), and determine where the function is increasing or decreasing.

Key Terms and Formulas

  • Quadratic function: (vertex form)

  • Vertex: is the maximum or minimum point

  • Roots: Values of where

  • Increasing/Decreasing: Intervals where the function rises or falls

Step-by-Step Guidance

  1. Identify the vertex by comparing the function to the vertex form .

  2. Determine if the parabola opens upwards or downwards by looking at the sign of .

  3. Find the roots by setting and solving for .

  4. Determine the interval where the function is increasing (to the left or right of the vertex) and where it is decreasing.

  5. Do not compute the exact roots or intervals yet; set up the equations needed to find them.

Try solving on your own before revealing the answer!

Final Answer:

Vertex: (maximum point)

Roots:

Increasing:

Decreasing:

The negative leading coefficient means the parabola opens downward, so the vertex is a maximum.

Q3. For , find the x-intercept, y-intercept, domain, and range.

Background

Topic: Linear Functions

This question checks your understanding of intercepts and the domain/range of a basic linear function.

Key Terms and Formulas

  • x-intercept: Set and solve for

  • y-intercept: Evaluate

  • Domain and range for linear functions

Step-by-Step Guidance

  1. Set and solve for to find the x-intercept.

  2. Substitute into to find the y-intercept.

  3. Recall that the domain and range of are all real numbers.

  4. Write the intercepts as ordered pairs, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

x-intercept:

y-intercept:

Domain:

Range:

This is a straight line through the origin with slope 2.

Q4. For , identify the parent function, range, transformations, intercepts, intervals of increase/decrease, and critical values.

Background

Topic: Quadratic Functions and Transformations

This question tests your ability to analyze a quadratic function, including its parent function, transformations, and key features.

Key Terms and Formulas

  • Parent function:

  • Transformation: Vertical stretch/compression, reflection, translation

  • Intercepts: Where the graph crosses the axes

  • Critical value: Vertex of the parabola

Step-by-Step Guidance

  1. Identify the parent function () and note the coefficient (or ).

  2. Describe the transformation: vertical stretch by a factor of .

  3. Find the y-intercept by evaluating .

  4. Determine the vertex (critical value) and whether the parabola opens up or down.

  5. Set up the process for finding intervals of increase and decrease, but do not compute the exact intervals yet.

Try solving on your own before revealing the answer!

Final Answer:

Parent function:

Range:

Transformation: Vertical stretch by

Intercept:

Increasing:

Decreasing:

Critical value: Vertex at

Q5. For , identify the parent function, range, transformations, intervals of increase/decrease, and critical values.

Background

Topic: Linear Functions and Transformations

This question asks you to analyze a linear function, including its parent function, transformations, and key features.

Key Terms and Formulas

  • Parent function:

  • Transformation: Reflection, translation, vertical/horizontal shifts

  • Critical value: For linear functions, this usually refers to intercepts

Step-by-Step Guidance

  1. Simplify the function by combining like terms.

  2. Identify the slope and y-intercept from the simplified equation.

  3. Describe the transformation(s) from the parent function .

  4. Determine the domain and range for the function.

  5. Set up the process for finding intervals of increase/decrease, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

The function simplifies to .

Parent function:

Range:

Transformation: Reflection over the x-axis

Increasing: Never

Decreasing: Always

Critical value: y-intercept at

Q6. For , identify the parent function, range, transformations, intervals of increase/decrease, and critical values.

Background

Topic: Linear Functions and Transformations

This question is similar to Q5, focusing on analyzing a linear function and its transformations.

Key Terms and Formulas

  • Parent function:

  • Transformation: Reflection, translation

  • Critical value: Intercepts

Step-by-Step Guidance

  1. Simplify the function by combining constants and like terms.

  2. Identify the slope and y-intercept from the simplified equation.

  3. Describe the transformation(s) from the parent function.

  4. Determine the domain and range for the function.

  5. Set up the process for finding intervals of increase/decrease, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

The function simplifies to .

Parent function:

Range:

Transformation: Reflection over the x-axis, shift down by 1

Increasing: Never

Decreasing: Always

Critical value: y-intercept at

Q7. For , identify the parent function, range, transformations, intervals of increase/decrease, and critical values.

Background

Topic: Linear Functions and Transformations

This question asks you to analyze a linear function with both a horizontal shift and a vertical shift.

Key Terms and Formulas

  • Parent function:

  • Transformation: Reflection, horizontal and vertical shifts

  • Critical value: Intercepts

Step-by-Step Guidance

  1. Distribute the and simplify the function.

  2. Identify the slope and y-intercept from the simplified equation.

  3. Describe the transformation(s) from the parent function.

  4. Determine the domain and range for the function.

  5. Set up the process for finding intervals of increase/decrease, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

The function simplifies to .

Parent function:

Range:

Transformation: Reflection over the x-axis, shift up by 1

Increasing: Never

Decreasing: Always

Critical value: y-intercept at

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