Q1. Find the equation of the line passing through the point (2, 5) with slope m = 3, in slope-intercept form.

Background

Topic: Equations of Lines

This question tests your ability to write the equation of a line given a point and a slope, and to express it in slope-intercept form ().

Key Terms and Formulas

• Slope-intercept form:

• Point-slope form:

• is a point on the line, is the slope.

Step-by-Step Guidance

1. Start with the point-slope form: , where and .

2. Substitute the given values into the formula: .

3. Expand the right side: .

4. Isolate to write the equation in slope-intercept form ().

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Q2. Find the equation of the line passing through the points (4, 0) and (0, 4), in slope-intercept form.

Background

Topic: Equations of Lines from Two Points

This question tests your ability to find the equation of a line given two points, using the slope formula and then writing the equation in slope-intercept form.

Key Terms and Formulas

• Slope formula:

• Slope-intercept form:

Step-by-Step Guidance

1. Label the points: and .

2. Calculate the slope using the formula: .

3. Simplify the slope expression.

4. Use the point-slope form with one of the points and the calculated slope.

5. Rearrange to slope-intercept form ().

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Q3. A line passes through the points (3, p) and (1, 4), and has slope m = 3. Find the value of p and the equation of this line in slope-intercept form.

Background

Topic: Finding Unknown Coordinates and Line Equations

This question tests your ability to use the slope formula to find an unknown coordinate, and then write the equation of the line in slope-intercept form.

Key Terms and Formulas

• Slope formula:

• Slope-intercept form:

Step-by-Step Guidance

1. Label the points: and .

2. Set up the slope equation: .

3. Solve for by simplifying and isolating $p$.

4. Once is found, use either point and the slope to write the equation in point-slope form.

5. Rearrange to slope-intercept form ().

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Q4. Find the equation of the vertical line passing through the point (5, 3).

Background

Topic: Vertical Lines

This question tests your understanding of the equation of a vertical line, which has an undefined slope and passes through a specific x-coordinate.

Key Terms and Formulas

• Vertical line equation: , where is the x-coordinate of the point.

Step-by-Step Guidance

1. Recall that a vertical line through has the equation .

2. Identify the x-coordinate from the given point .

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Q5. Find the equation of the horizontal line passing through the point (5, 3).

Background

Topic: Horizontal Lines

This question tests your understanding of the equation of a horizontal line, which has a slope of zero and passes through a specific y-coordinate.

Key Terms and Formulas

• Horizontal line equation: , where is the y-coordinate of the point.

Step-by-Step Guidance

1. Recall that a horizontal line through has the equation .

2. Identify the y-coordinate from the given point .

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Q6. Show that is an even function.

Background

Topic: Even and Odd Functions

This question tests your ability to determine if a function is even by checking if for all in the domain.

Key Terms and Formulas

• Even function: for all in the domain.

Step-by-Step Guidance

1. Write the definition: .

2. Compute : Substitute into the function.

3. Simplify and compare it to .

Try solving on your own before revealing the answer!

Q7. Show that is an odd function.

Background

Topic: Even and Odd Functions

This question tests your ability to determine if a function is odd by checking if for all in the domain.

Key Terms and Formulas

• Odd function: for all in the domain.

Step-by-Step Guidance

1. Write the definition: .

2. Compute : Substitute into the function.

3. Simplify and compare it to .

Try solving on your own before revealing the answer!