Q2. Use the graph described above to determine the following values:

Background

Topic: Reading Values from a Function's Graph

This question tests your ability to interpret a function's graph and extract specific values for at given values, as well as to find values for which equals a specified number.

Key Terms:

• Function notation: represents the output of the function for input .

• Graph interpretation: The -value at a given is .

Step-by-Step Guidance

1. Locate on the graph and find the corresponding -value. This is .

2. Locate on the graph and find the corresponding -value. This is .

3. To find where , look for points on the graph where the -value is and note the corresponding values.

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Q3. Using the same graph from Question 2, identify the boundaries of the function.

Background

Topic: Domain and Range of a Function

This question tests your understanding of how to determine the domain (possible values) and range (possible values) from a graph.

Key Terms:

• Domain: The set of all values for which the function is defined (horizontal extent).

• Range: The set of all values the function takes (vertical extent).

Step-by-Step Guidance

1. Examine the graph from left to right to determine the smallest and largest values where the graph exists. These are the domain boundaries.

2. Examine the graph from bottom to top to determine the lowest and highest values the graph reaches. These are the range boundaries.

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Q4. Given , find all intercepts algebraically.

Background

Topic: Algebraic Intercepts of Functions

This question tests your ability to find -intercepts (where the graph crosses the -axis) and -intercepts (where the graph crosses the -axis) using algebraic methods.

Key Terms and Formulas:

• -intercept: Set and solve .

• -intercept: Set and solve for .

Step-by-Step Guidance

1. To find -intercepts, set and solve .

2. Factor or use the quadratic formula to solve for .

3. To find the -intercept, substitute into and compute .

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Q5. Identify if the following graph descriptions represent an even, odd, or neither function:

Background

Topic: Symmetry and Parity of Functions

This question tests your understanding of even and odd functions based on their symmetry properties.

Key Terms:

• Even function: Symmetric with respect to the -axis ().

• Odd function: Symmetric with respect to the origin ().

Step-by-Step Guidance

1. For the "U-shaped" parabola with vertex at , check if the graph is symmetric about the -axis.

2. For the line passing through the origin with points and , check if the graph is symmetric about the origin.

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Q6. Test the function algebraically for parity.

Background

Topic: Algebraic Parity Test

This question tests your ability to determine whether a function is even, odd, or neither using algebraic substitution.

Key Terms and Formulas:

• Parity test: Substitute with in and compare the result to and .

Step-by-Step Guidance

1. Replace with in the function: .

2. Simplify the expression for .

3. Compare to and to determine parity.

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Q7. Using the graph, describe the intervals where is increasing, decreasing, or constant.

Background

Topic: Increasing, Decreasing, and Constant Intervals

This question tests your ability to analyze a graph and identify intervals of monotonicity (where the function goes up, down, or stays flat).

Key Terms:

• Increasing interval: gets larger as increases.

• Decreasing interval: gets smaller as increases.

• Constant interval: stays the same as increases.

Step-by-Step Guidance

1. Look at the graph and identify segments where the function rises, falls, or remains flat as you move from left to right.

2. Use the -values to describe each interval.

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Q8. Using the graph from Question 7, identify the local minima and maxima.

Background

Topic: Local Extrema

This question tests your ability to find "peaks" (local maxima) and "valleys" (local minima) on a graph.

Key Terms:

• Local maximum: Highest point in a neighborhood.

• Local minimum: Lowest point in a neighborhood.

Step-by-Step Guidance

1. Examine the graph for points where the function changes from increasing to decreasing (local maxima) or decreasing to increasing (local minima).

2. Identify the -values and corresponding -values for these points.

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Q9. Does the graph in Question 7 have an absolute maximum? If so, what is the highest -value?

Background

Topic: Absolute Extrema

This question tests your ability to identify the highest value a function attains on its entire domain.

Key Terms:

• Absolute maximum: The largest -value on the graph.

Step-by-Step Guidance

1. Scan the graph for the highest point (largest -value).

2. Record the -value and the corresponding -value.

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Q10. Find the average rate of change for from to .

Background

Topic: Average Rate of Change (AROC)

This question tests your ability to calculate the average rate of change (slope) between two points on a function.

Key Formula:

• Average Rate of Change:

Step-by-Step Guidance

1. Compute and using .

2. Plug these values into the formula .

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Q11. What is the slope of the secant line containing the points and ?

Background

Topic: Secant Lines

This question tests your understanding of the geometric meaning of average rate of change as the slope of a secant line between two points.

Key Terms:

• Secant line: A line connecting two points on a curve.

• Slope: between two points.

Step-by-Step Guidance

1. Recall that the slope of the secant line is the same as the average rate of change calculated in Question 10.

2. Use the formula .

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Q12. Find the average rate of change for from $1x$.

Background

Topic: Average Rate of Change (Generalized)

This question tests your ability to use the average rate of change formula for a variable interval and simplify the result.

Key Formula:

• Average Rate of Change:

Step-by-Step Guidance

1. Compute for .

2. Write the expression .

3. Simplify the numerator by factoring, if possible.

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