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Intermediate Algebra Study Guide: Linear Equations, Functions, and Graphs

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Q1. Is \( \left( -\frac{2}{3}, 8 \right) \) a solution to \( 9x + y = 2 \)?

Background

Topic: Solutions to Linear Equations

This question tests your ability to determine if a given point is a solution to a linear equation by substituting the values into the equation.

Key Terms and Formulas:

  • Linear equation: An equation of the form \( ax + by = c \).

  • Solution: A point \( (x, y) \) that makes the equation true when substituted.

Step-by-Step Guidance

  1. Identify the values to substitute: \( x = -\frac{2}{3} \), \( y = 8 \).

  2. Substitute these values into the equation: \( 9x + y = 2 \).

  3. Calculate \( 9 \times \left(-\frac{2}{3}\right) + 8 \).

  4. Check if the result equals 2.

Try solving on your own before revealing the answer!

Q2. Find the x- and y-intercepts of \( -5x + 3y = -30 \)

Background

Topic: Intercepts of Linear Equations

This question asks you to find where the line crosses the x-axis (y = 0) and y-axis (x = 0).

Key Terms and Formulas:

  • x-intercept: Set \( y = 0 \) and solve for \( x \).

  • y-intercept: Set \( x = 0 \) and solve for \( y \).

Step-by-Step Guidance

  1. For the x-intercept, set \( y = 0 \) in the equation and solve for \( x \).

  2. For the y-intercept, set \( x = 0 \) in the equation and solve for \( y \).

  3. Solve each equation to find the intercepts.

Try solving on your own before revealing the answer!

Q3. Graph \( x = 4 \)

Background

Topic: Graphing Vertical and Horizontal Lines

This question tests your understanding of how to graph a vertical line on the coordinate plane.

Key Terms and Formulas:

  • Vertical line: All points where \( x \) is constant.

Step-by-Step Guidance

  1. Recognize that \( x = 4 \) is a vertical line passing through all points where \( x = 4 \).

  2. Draw a straight line parallel to the y-axis at \( x = 4 \).

Try solving on your own before revealing the answer!

Q4. What is the slope of the line through (2, 1) and (8, 4)?

Background

Topic: Slope of a Line

This question asks you to calculate the slope between two points using the slope formula.

Key Terms and Formulas:

  • Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Step-by-Step Guidance

  1. Label the points: \( (x_1, y_1) = (2, 1) \), \( (x_2, y_2) = (8, 4) \).

  2. Substitute the values into the slope formula.

  3. Calculate the numerator \( y_2 - y_1 \) and denominator \( x_2 - x_1 \).

Try solving on your own before revealing the answer!

Q5. Find the slope of the line through (–3, –7) and (–11, –11)

Background

Topic: Slope of a Line

This question is similar to Q4, focusing on calculating the slope between two points.

Key Terms and Formulas:

  • Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Step-by-Step Guidance

  1. Label the points: \( (x_1, y_1) = (–3, –7) \), \( (x_2, y_2) = (–11, –11) \).

  2. Substitute the values into the slope formula.

  3. Calculate the numerator and denominator.

Try solving on your own before revealing the answer!

Q6. In slope-intercept form, find the line through (4, 6) and (0, –2)

Background

Topic: Equation of a Line in Slope-Intercept Form

This question asks you to find the equation of a line given two points, using the slope-intercept form \( y = mx + b \).

Key Terms and Formulas:

  • Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

  • Slope-intercept form: \( y = mx + b \)

Step-by-Step Guidance

  1. Find the slope \( m \) using the two points.

  2. Use one point and the slope to solve for \( b \) in \( y = mx + b \).

  3. Write the equation in slope-intercept form.

Try solving on your own before revealing the answer!

Q7. Graph the line through (1, 2) with slope \( \frac{3}{5} \)

Background

Topic: Graphing Lines Using Point and Slope

This question tests your ability to graph a line given a point and a slope.

Key Terms and Formulas:

  • Point-slope form: \( y - y_1 = m(x - x_1) \)

Step-by-Step Guidance

  1. Start at the point (1, 2) on the graph.

  2. Use the slope \( \frac{3}{5} \) to find another point: rise 3 units, run 5 units.

  3. Draw the line through these points.

Try solving on your own before revealing the answer!

Q8. In slope-intercept form, what is the equation of the line with slope 3 through the point (2, 5)?

Background

Topic: Equation of a Line in Slope-Intercept Form

This question asks you to write the equation of a line given a slope and a point.

Key Terms and Formulas:

  • Slope-intercept form: \( y = mx + b \)

Step-by-Step Guidance

  1. Substitute the slope \( m = 3 \) and the point (2, 5) into \( y = mx + b \).

  2. Solve for \( b \).

  3. Write the final equation in slope-intercept form.

Try solving on your own before revealing the answer!

Q9. In slope-intercept form, what is the equation of the line through the points (3, –6) and (–6, 3)?

Background

Topic: Equation of a Line in Slope-Intercept Form

This question asks you to find the equation of a line given two points.

Key Terms and Formulas:

  • Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

  • Slope-intercept form: \( y = mx + b \)

Step-by-Step Guidance

  1. Calculate the slope \( m \) using the two points.

  2. Use one point and the slope to solve for \( b \).

  3. Write the equation in slope-intercept form.

Try solving on your own before revealing the answer!

Q10. What is the equation of a horizontal line through (2, –5)?

Background

Topic: Horizontal and Vertical Lines

This question tests your understanding of the equation of a horizontal line, which has a constant y-value.

Key Terms and Formulas:

  • Horizontal line: \( y = c \), where \( c \) is the y-value of the point.

Step-by-Step Guidance

  1. Identify the y-value of the given point (2, –5).

  2. Write the equation as \( y = \text{(that y-value)} \).

Try solving on your own before revealing the answer!

Q11. For the function \( q(x) = \frac{2x - 3}{x + 1} \), find \( q(4) \), \( q(0) \), and \( q(-1) \)

Background

Topic: Evaluating Rational Functions

This question asks you to substitute specific values into a rational function and simplify.

Key Terms and Formulas:

  • Rational function: A function of the form \( \frac{P(x)}{Q(x)} \).

Step-by-Step Guidance

  1. Substitute \( x = 4 \) into \( q(x) \) and simplify.

  2. Substitute \( x = 0 \) into \( q(x) \) and simplify.

  3. Substitute \( x = -1 \) into \( q(x) \) and check if the denominator is zero.

Try solving on your own before revealing the answer!

Q12. Find all values of \( x \) such that \( f(x) = -5 \)

Background

Topic: Solving Quadratic Equations Graphically

This question asks you to find the x-values where the graph of \( f(x) \) intersects the line \( y = -5 \).

Key Terms and Formulas:

  • Intersection points: Where the graph meets \( y = -5 \).

Step-by-Step Guidance

  1. Look at the graph and find the x-values where the curve crosses \( y = -5 \).

  2. Write down the corresponding x-values.

Try solving on your own before revealing the answer!

Graph of quadratic function for Q12

Q13. For the function graphed below, state the domain and range in interval notation.

Background

Topic: Domain and Range of a Function

This question asks you to determine the set of all possible x-values (domain) and y-values (range) for the given graph.

Key Terms and Formulas:

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

  • Range: All y-values the function attains.

Step-by-Step Guidance

  1. Examine the graph to see the leftmost and rightmost x-values (domain).

  2. Identify the lowest and highest y-values (range).

  3. Express both in interval notation.

Try solving on your own before revealing the answer!

Graph for domain and range question Q13

Q14. Graph the piecewise defined function \( f(x) = \begin{cases} x & x < 0 \\ -x + 2 & x \geq 0 \end{cases} \)

Background

Topic: Piecewise Functions

This question asks you to graph a function defined by two different expressions depending on the value of x.

Key Terms and Formulas:

  • Piecewise function: A function defined by different expressions for different intervals of x.

Step-by-Step Guidance

  1. For \( x < 0 \), graph \( f(x) = x \) (a line through the origin with slope 1, but only for x less than 0).

  2. For \( x \geq 0 \), graph \( f(x) = -x + 2 \) (a line with slope -1 and y-intercept 2, for x greater than or equal to 0).

  3. Make sure to indicate open or closed circles at the transition point (x = 0) as appropriate.

Try solving on your own before revealing the answer!

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