Algebra B Midterm Study Guide

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Questions and Answers

The number of people, f(x), at a tourist destination is a function of time, in months. Using the graph of f(x), find f(4).

5

Select the transformations of h(x) = -3|x - 4| + 5 as it relates to the parent function. Select all that apply. (Note: Parent function - f(x) = |x|)

  • Translated right 4 units (correct)
  • Vertical stretch by a factor of 3 (correct)
  • Reflection over x-axis (correct)
  • Translated up 5 units (correct)
  • Translated left 4 units

For f(x) = 3x + 2 and h(x) = x^2 -1, find the value of 4[f(-2) + h(3)]. (Hint: Recall that the number in parentheses is what you plug in for x.)

16

For f(x) = -2x + 3, find the value of f(3) - f(4).

<p>-3</p> Signup and view all the answers

Find the range of the function below if the domain is {-2, 1, 3}. (Note: Domain = x-values)

<p>{-15, -3, 25} (C)</p> Signup and view all the answers

The distance, f(x), from a house to a cafe is a function of time. Using the graph of f(x), find f(7.5).

<p>7.5</p> Signup and view all the answers

Select the transformations of k(x) = 2|-x + 3| - 4 as it relates to the parent function. Select all that apply.

<p>Vertical stretch by a factor of 2 (B), Translated down 4 units (D), Reflected over the y-axis (G), Translated left 3 units (H)</p> Signup and view all the answers

Write an equation for the nth term of the arithmetic sequence 10, 8, 6, 4, ... (Note: a_n = a_1 + (n - 1)d).

<p>-2n + 12</p> Signup and view all the answers

What is the slope of the line that passes through (-3, 5) and (3, 6)?

<p>1/6 (C)</p> Signup and view all the answers

Which equation models the line on the graph? (Note: Use 2 points to find slope. Use where the line crosses the y-axis to find the y-intercept.)

<p>y = 1/2x + 3 (D)</p> Signup and view all the answers

Find the value of n so that the line through the points has the given slope. (4, 6) and (n, -2) and slope = 1/2

<p>-12</p> Signup and view all the answers

Which equation models the line on the graph? (Note: These are in point-slope form! Find the slope and use one point to create the equation.)

<p>y - 7 = -3/4(x + 0) (D)</p> Signup and view all the answers

Sopha owns a cafe on the verge of shutting down. She charges $7.50 for 3 pancakes and a cup of coffee. She charges $3.00 for one pancake and a cup of coffee. How much does Sopha charge for a pancake? How much does she charge for a cup of coffee?

<p>The cost of a pancake is $2.25, and the cost of a cup of coffee is $0.75.</p> Signup and view all the answers

BrightTalk Mobile charges $0.25 per minute plus a monthly base fee. In August, Sarah used 150 minutes, and her bill was $45. Write and solve an equation to find the cost of the monthly base fee.

<p>The monthly base fee is $7.50.</p> Signup and view all the answers

Use your equation to find the cost if Sarah uses 180 minutes in September.

<p>The cost in September will be $52.50.</p> Signup and view all the answers

Which equation represents a line passing through the point (4, 2) with a slope of -2?

<p>y = -2x + 30 (D)</p> Signup and view all the answers

Write the equation of the line that passes through (-6, -2) and is perpendicular to y = 3/2x - 5.

<p>y = -2/3x - 6 (A)</p> Signup and view all the answers

Which equation models the line graphed below in point-slope form? (Hint: y - y1 = m(x - x1))

<p>y + 2 = 2/3(x - 3) (A)</p> Signup and view all the answers

Determine the values of A, B, and C when y - 3 = 4(x + 2) is written in standard form. Show your work.

<p>A = 4, B = -1, C = -11</p> Signup and view all the answers

The scatter plot shows the average price of a National Football League ticket from 2007 to 2016.

a) Use the points (2007, 67.11) and (2016, 92.98) to write an equation of the line of fit in slope-intercept form. Let x be the number of years since 2006. Show your work. b) If the trend continues, what will be the price of a ticket in 2030?

<p>a) <em>y = 2.87x + 62.75</em> b) $131.63</p> Signup and view all the answers

Solve and graph -2y > 10.

<p>y &lt; -5</p> Signup and view all the answers

Which inequalities have no solution? Select all that apply.

<p>4y + 3 ≤ 2(2y - 2) (D)</p> Signup and view all the answers

Javier is taking part in a charity event at his school. Candy is being sold for $3 per bag and cookies are being sold for $6 per bag. What inequality represents the number of bags of candies (x) and the number of bags of cookies (y) that need to be sold to raise at least $600?

<p>3x + 6y ≥ 600 (C)</p> Signup and view all the answers

Solve -8y < -3y + 5 for y.

<p>y &gt; -1</p> Signup and view all the answers

Select the possible solutions of 15x ≥ 10 + 12x. Select ALL that apply.

<p>x ≥ 10/3 (A), x ≥ 2 (B), 10 ≤ x (C), x ≥ 3 (D)</p> Signup and view all the answers

Which graph shows 5x ≥ 20 or -x - 14 < -16?

<p>-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 (A), -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 (B), -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 (C), -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 (D)</p> Signup and view all the answers

Select the possible solutions of 8z - 4 ≤ 3z + 7. Select ALL that apply.

<p>2 ≤ z (B), z ≤ 11/5 (C), 11/5 ≤ z (E)</p> Signup and view all the answers

Which graph shows the solution for 1 < 3x - 2 < 10?

<p>1&lt;x AND x ≤ 4 (C)</p> Signup and view all the answers

Which graph shows 2(3x - 7) < -14 or 3x + 10 > 13?

<p>x &lt; 0 or x &gt; 7 (D)</p> Signup and view all the answers

Solve and graph 8(-x + 1) ≥ -65 and 9 - 8x > -79 on a number line.

<p>x ≤ 11 and x ≤ 9.75</p> Signup and view all the answers

A thermometer is accurate within 0.3 degrees Fahrenheit. a) Write an absolute value inequality that represents the possible actual temperature t if the thermometer reads 68.5°F. b) Solve the inequality and graph it on a number line.

<p>a) <em>|t - 68.5| ≤ 0.3</em> b) 68.2 ≤ <em>t</em> ≤ 68.8</p> Signup and view all the answers

Solve the inequality |2y + 5| < 15. Then, graph your solution.

<p>-10 &lt; <em>y</em> &lt; 5</p> Signup and view all the answers

Graph the inequality 3/2x + 5 - y < 6. Give one point that is a solution.

<p>(0, 0)</p> Signup and view all the answers

Match each of the 3 graphs to its inequality i) 3x + 6y > -18 ii) 6x + 6y < -18 iii) 6x - 6y > -18

For which graph is (7, 0) a solution?

<p>Graph A (A), Graph C (B)</p> Signup and view all the answers

Solve and graph the inequality |4x - 7| >11.

<p>x &lt; -1 or x &gt; 4.5</p> Signup and view all the answers

Which of the following represents a line parallel to the x-axis?

<p>y = 9 (B)</p> Signup and view all the answers

Select all the ways the system of equations could be solved.

8x - 9y = 13 16x + 3y = 22

<p>Multiply the second equation by 2, then add the equations. (A), Multiply the first equation by 2, then subtract the equations. (C), Multiply the second equation by 3, then subtract the equations. (E)</p> Signup and view all the answers

Solve the system of equations using substitution.

c - d = 6 c = 4d - 8

<p>(<em>c</em>, <em>d</em>) = (24, 8)</p> Signup and view all the answers

Solve the system of equations using elimination.

1/2 e - f = 7 1/2 e + 5/2 f = 9

<p>(<em>e</em>, <em>f</em>) = (-15, 8)</p> Signup and view all the answers

What is the value of x if Lines 1 and 2 are parallel?

Line 1: (1, 3) and (4, 7) Line 2: (x, 10) and (2, 6)

<p>x = 5</p> Signup and view all the answers

Graph the system of inequalities. y ≥ 3x + 2 y < -x - 2

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Sketch the solution to the system of inequalities on the graph.

x + y < 2 4x + y < -1

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Sketch the solution to the system of inequalities on the graph.

x ≥ -2 2x - 3y > 3

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What is the value of x if Lines 1 and 2 are perpendicular?

Line 1: (3, 5) and (6, 9) Line 2: (x, 4) and (5, 2)

<p>x = 5</p> Signup and view all the answers

Some really old and really homemade cars are having a race. Car A is 10 feet behind the starting line and moves at a rate of 6 feet per second. Car B has cheated and is 25 feet ahead of the starting line. Car B also moves at a rate of 4 feet per second. The two racers of the cars want to know how many seconds, t, will pass before the two cars are at the same distance. a) Write a system of equations that can be used to represent the situation. b) Solve the system to figure out how many seconds will pass before the cars are the same distance from the starting line.

<p>a) <em>d_1</em> = 6<em>t</em> - 10 and <em>d_2</em> = 4<em>t</em> + 25 b) <em>t</em> = 17.5 seconds</p> Signup and view all the answers

Flashcards

find f(4)

The value of a function at a specific input (x). In this case, find the output (y) of the function when x = 4.

Transformations of ℎ(𝑥) = -3|𝑥 - 4| + 5

The transformations of a function compared to its parent function. In this case, identify how ℎ(𝑥) = -3|𝑥 - 4| + 5 differs from f(x) = |x|.

Function of People at a Tourist Destination

The number of people at a tourist destination, f(x), is a function of the time in months, x. The graph of f(x) shows this relationship.

Parent Function

A parent function is the simplest form of a function, like f(x) = |x|. Transformations are applied to create variations, making the function more complex.

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Composite Function

To find a composite function, you evaluate the inner function first, then use that output as the input for the outer function.

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Find 4[f(-2) + h(3)]

The value of a function at a specific input (x). Find the output (y) of the function when x = -2 and x = 3, then plug the results into the expression.

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Find f(3) - f(4)

The value of a function at a specific input (x). Calculate the outputs for both x = 3 and x = 4, then subtract the results.

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Range of a Function

The set of all possible y-values of a function. To find it, plug in the given x-values and calculate the corresponding y-values.

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Function of Distance from House to Cafe

The distance, f(x), from a house to a cafe is a function of the time in x. The graph of f(x) shows this relationship.

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Transformations of 𝑘(𝑥) = 2 3| − 4

The transformations of a function compared to its parent function. In this case, identify how 𝑘(𝑥) = 2 3| − 4 differs from f(x) = |x|.

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nth Term of Arithmetic Sequence

An arithmetic sequence is a pattern where the difference between consecutive terms is constant. The nth term is a formula to find any term based on the first term and the constant difference.

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Slope of a Line

The slope of a line measures its steepness or slant. It's the rate of change of y-values (vertical) for every change in x-values (horizontal).

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Slope of a Horizontal Line

A horizontal line's slope is always zero (no change in y for any change in x).

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Slope of a Vertical Line

A vertical line's slope is undefined. This is because there is infinite change in y for no change in x.

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Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b. Here, m is the slope and b is the y-intercept.

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Finding Slope and Y-Intercept

Using two points on a line, the slope can be calculated using the formula y2−y1. The y-intercept is where the line crosses the y-axis.

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Finding Value of n

The slope of a line is the rate of change of y for every change in x. Use the formula y2−y1 and plug in the coordinates, solve for n.

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Point-Slope Form

Point-slope form of a linear equation is y - y1 = m(x - x1). Find the slope (m) and use one of the points (x1, y1).

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Solving System of Equations for Prices

Set up a system of equations and solve for the variables. The solution will be the price of each item.

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Linear Equation

A linear equation is a relationship between x and y that can be represented by a straight line. It describes a constant rate of change.

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Equation of Line in Point-Slope Form

Point-slope form of a linear equation is y - y1 = m(x - x1). Use the given point and slope to write the equation.

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Equation of Perpendicular Line

A line perpendicular to another line has a slope that is the negative reciprocal of the original line's slope. Find the negative reciprocal of the given slope.

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Equation of Line from Graph

Point-slope form of a linear equation is y - y1 = m(x - x1). Find the slope from the graph and use a point to write the equation.

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Standard Form of a Linear Equation

Rewrite the equation in standard form by isolating y and moving the x term to the left side.

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Equation of Line of Fit

Use two points (x, y) to calculate the slope and y-intercept of the scatter plot. Then, write the equation in slope-intercept form.

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Predicting Values Using Line of Fit

Use the equation of the line of fit to predict the output (y) at a specific input (x).

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Solving Linear Inequalities

An inequality shows the relationship between values that are not equal. The solution involves finding the values that satisfy the inequality.

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Inequalities with No Solution

Inequalities that have no solution are those where the variables cancel out and the resulting statement is false.

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Inequality for Target Amount

An inequality representing the number of items to be sold (x and y) to reach a target amount or greater.

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Solving Inequalities for y

Solve the inequality by isolating y on one side. Simplify and remember to reverse the inequality sign if multiplying or dividing by a negative number.

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Possible Solutions for Inequality

Find the values that satisfy the inequality. The solution will be a range of values that satisfy the given inequality.

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Graphing Solution to Inequality

Graph the solutions to the inequality on a number line. The direction of the inequality sign determines the type of boundary.

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Solving Absolute Value Inequalities

The absolute value of a number is its distance from zero. Solve the inequality by considering both positive and negative cases.

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Graphing Solution for Absolute Value Inequality

Graph the solutions to the absolute value inequality on a number line. The inequality determines the type of boundary.

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Graphing Linear Inequalities

Graph the inequality by plotting a line based on the equation. The shading indicates the solution region based on the inequality sign.

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Line Parallel to X-Axis

A horizontal line is a straight line that never changes its y-value, making its slope zero.

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Solving Systems of Equations

A system of equations can be solved using several methods, including elimination, substitution, or graphing.

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Solving Systems Using Substitution

Substitution involves solving one equation for one variable and substituting it into the other equation. Solve for one variable, then substitute.

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Solving Systems Using Elimination

Elimination involves multiplying equations by appropriate factors to eliminate one variable. Multiply, add, solve for one variable.

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Parallel Lines

Parallel lines have the same slope. Calculate the slope of line 1, and set it equal to the slope of line 2 to find the value of x.

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Graphing Systems of Inequalities

Graph the system of inequalities. The solution region is where the shaded areas of each inequality overlap. The solution is the region where all the inequalities are satisfied.

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Study Notes

Algebra B Midterm Study Guide

  • Module 3: Relations and Functions
    • Problem 1: Given a graph of a function representing the number of people at a tourist destination as a function of time (in months). Find f(4).
      • f(4) = 5 (The number of people at the destination when the time is 4 months.)
    • Problem 2: Identify transformations from the parent function for h(x) = -3|x - 4| + 5.
      • Reflection over x-axis
      • Vertical stretch by a factor of 3
      • Translated right 4 units
      • Translated up 5 units
      • Parent Function f(x) = |x|
    • Problem 3: Evaluate 4[f(-2) + h(3)] for f(x) = 3x + 2 and h(x) = x² - 1.
      • f(-2) = -4
      • h(3) = 8
      • 4[-4 + 8] = 16
    • Problem 4: Find the value of f(3) - f(4) for f(x) = -2x + 3.
      • f(3) = -3
      • f(4) = -5
      • -3 - (-5) = 2

Module 3: Relations and Functions (Continued)

  • Problem 5: For the given function, find the range if the domain is {-2, 1, 3}.
    • f(x) = x³ + x - 5/
      • f(-2) = -15
      • f(1) = -3
      • f(3) = 25
      • Range: {-15, -3, 25}
  • Problem 6: Find the value of f(7.5) from a graph of the distance from a house to a cafe as a function of time.
    • f(7.5) = 7.5
  • Problem 7: Identify transformations of k(x) = 2|-x + 3| - 4 from the parent function.
    • Reflection over the y-axis
    • Horizontal translation left 3 units
    • Vertical stretch by a factor of 2
    • Vertical translation down 4 units -Parent Function:f(x)=|x|

Module 4: Linear and Nonlinear Functions

  • Problem 8: Find the nth term of the arithmetic sequence 10, 8, 6, 4.
    • a₄ = a₁ + (n-1)d
    • an = -2n + 12
  • Problem 9: Find the slope of a line passing through (-3, 5) and (3, 6).
    • m = (6 - 5) / (3 - (-3)) = 1/6
  • Problem 10: Find the slope of a line passing through (-3, 2) and (7, 2).
    • m = (2-2)/(7 - (-3)) = 0

Module 4: Linear and Nonlinear Functions (Continued)

  • Problem 11: Find the equation of the line in the given graph.
    • The line passes through (0, 3) and (2, 4).
    • m = 1/2
    • b = 3
    • y = 1/2 x + 3

Module 5: Creating Linear Equations

  • Problem 16: BrightTalk Mobile charges $0.25 per minute plus a monthly base fee. Sarah’s bill for 150 minutes was $45.
  • a) Find the monthly base fee. - Monthly base fee: $7.50
  • b) Find the bill if she uses 180 minutes in September
    • $52.50
  • Problem 17: Find the equation of a line passing through (4, 2) with slope of 1/2. -y = (1/2)x + (1/2)or y = (1/2)x - 1)

Module 6: Linear Inequalities

  • Problem 24: Solve $-2y > 10$

    • y < -5

  • Problem 25: Find inequalities with no solution

  • Problem 26: Find inequality for number of bags of candy (x) and cookies (y) sold to raise $600 or more. -3x + 6y ≥ 600

  • Problem 27: Solve -8y ≤ -3y + 5 for y

    • y ≥ -1

  • Problem 28: Select potential solutions to 15x ≥ 10 + 12x

  • Problem 29: Identify graph solution to 5x ≥ or or -x-14 <-16

  • Problem 30: Select correct potential solutions to 8(-4) ≤ 3z + 7

  • Problem 31: Find graph of solution to 1 ≤ 3x − 2 < 10

  • Problem 32: Find graph of solution 2(3x - 7) < -14 or 3x + 10 > 13

Module 7: Systems of Equations and Inequalities

  • Problem 39: Find a horizontal line among given equations
    • y = 9
  • Problem 40: Select correct ways to solve system of equations
  • Problem 41: Solve system of equations through substitution
  • Problem 42: Solve system of equations through elimination

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