Q4. Without sketching the graph, find the x-intercepts and y-intercepts of the equation 4x + 6y = 120.

Background

Topic: Linear Equations and Intercepts

This question tests your ability to find the x- and y-intercepts of a linear equation by setting one variable to zero and solving for the other.

Key Terms and Formulas

• x-intercept: The point(s) where the graph crosses the x-axis (y = 0).

• y-intercept: The point(s) where the graph crosses the y-axis (x = 0).

Step-by-Step Guidance

1. To find the x-intercept(s), set in the equation and solve for .

2. To find the y-intercept(s), set in the equation and solve for .

3. Write your answers as points: for x-intercepts and for y-intercepts.

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Q5. Evaluate the function at the given values and simplify.

Background

Topic: Function Evaluation

This question tests your ability to substitute values into a quadratic function and simplify the result.

Key Terms and Formulas

• Function evaluation: Substitute the given value for in .

• Quadratic function:

Step-by-Step Guidance

1. For , substitute into : .

2. For , substitute for : .

3. For , substitute for : .

4. Simplify each expression step by step, combining like terms.

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Q6. Use the graph to determine intervals where the function is increasing, decreasing, or constant.

Background

Topic: Intervals of Increase/Decrease/Constancy

This question tests your ability to read a graph and identify where the function rises, falls, or remains unchanged.

Key Terms

• Increasing interval: Where the graph goes up as increases.

• Decreasing interval: Where the graph goes down as increases.

• Constant interval: Where the graph is flat (horizontal).

Step-by-Step Guidance

1. Look at the graph and identify the -intervals where the function rises (increasing).

2. Identify the -intervals where the function falls (decreasing).

3. Check for any intervals where the function is constant (horizontal line).

4. Write the intervals using proper notation, such as for increasing, etc.

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Q7. Find the domain of the function .

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain of a rational function by identifying values that make the denominator zero.

Key Terms and Formulas

• Domain: All real -values for which the function is defined.

• Rational function: A function of the form .

Step-by-Step Guidance

1. Set the denominator equal to zero and solve for .

2. Find the values of that make the denominator zero; these must be excluded from the domain.

3. Write the domain in interval notation, excluding the problematic values.

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Q8. Graph the function

Background

Topic: Piecewise Functions

This question tests your ability to interpret and graph a piecewise-defined function.

Key Terms

• Piecewise function: A function defined by different expressions on different intervals.

Step-by-Step Guidance

1. For , graph the line only for .

2. For , graph the constant function for .

3. Pay attention to open and closed circles at to indicate which part of the function includes .

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Q10. Using the given graph, determine whether the function is even, odd, or neither.

Background

Topic: Symmetry of Functions

This question tests your ability to determine the symmetry of a function from its graph.

Key Terms

• Even function: Symmetric about the y-axis ().

• Odd function: Symmetric about the origin ().

• Neither: If the function does not have either symmetry.

Step-by-Step Guidance

1. Check if the graph is symmetric about the y-axis (mirror image left and right).

2. Check if the graph is symmetric about the origin (rotational symmetry).

3. If neither symmetry is present, the function is neither even nor odd.

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Q11. Find the slope of the line shown on the graph.

Background

Topic: Slope of a Line

This question tests your ability to calculate the slope of a line given two points on its graph.

Key Terms and Formulas

• Slope formula:

Step-by-Step Guidance

1. Identify two points on the line from the graph, such as and .

2. Plug the coordinates into the slope formula: .

3. Simplify the fraction to find the slope.

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Q15. Let . Find , , and . Sketch the graph of .

Background

Topic: Piecewise Functions and Graphing

This question tests your ability to evaluate and graph a piecewise function.

Key Terms

• Piecewise function: Defined by different expressions for different intervals of .

Step-by-Step Guidance

1. For , since , use the second piece: .

2. For , since , use the first piece: .

3. For , since , use the first piece: .

4. To sketch the graph, plot for and for .

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Q20. Find the equation of the line passing through the points and . Write in slope-intercept form.

Background

Topic: Equation of a Line

This question tests your ability to find the equation of a line given two points.

Key Terms and Formulas

• Slope formula:

• Slope-intercept form:

Step-by-Step Guidance

1. Calculate the slope using the two points: .

2. Use the point-slope form: , then solve for .

3. Substitute one of the points and the slope to find the y-intercept .

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Q3. Graph by starting with a library function and applying shifting and reflecting techniques.

Background

Topic: Transformations of Functions

This question tests your understanding of how to graph quadratic functions using transformations.

Key Terms and Formulas

• Library function:

• Transformation: Shifting, reflecting, and vertical translation.

Step-by-Step Guidance

1. Start with the basic parabola .

2. Apply a horizontal shift left by 1 unit: .

3. Reflect the graph over the x-axis: .

4. Shift the graph up by 4 units: .

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Q4. Write an equation for a function whose graph fits the following description: The graph of is shifted 7 units right, reflected about the x-axis, and shifted 9 units down.

Background

Topic: Transformations of Radical Functions

This question tests your ability to write equations for transformed functions.

Key Terms and Formulas

• Horizontal shift: shifts right by units.

• Reflection about x-axis: .

• Vertical shift: shifts up/down by units.

Step-by-Step Guidance

1. Shift right by 7: .

2. Reflect about the x-axis: .

3. Shift down by 9: .

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Q12. Graph given the graph of .

Background

Topic: Transformations—Horizontal Compression

This question tests your ability to apply horizontal compression to a function.

Key Terms and Formulas

• Horizontal compression: compresses the graph horizontally by a factor of .

Step-by-Step Guidance

1. For , every -value on the graph of is replaced by .

2. This means the graph is compressed horizontally by a factor of .

3. Match the compressed graph to the correct option among the choices.

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