Podcast
Questions and Answers
What is the vertex form of the equation y = -6x² - 24x - 31?
What is the vertex form of the equation y = -6x² - 24x - 31?
y = -6(x + 2)² - 7
What are the coordinates of the vertex for the equation y = -6x² - 24x - 31?
What are the coordinates of the vertex for the equation y = -6x² - 24x - 31?
(-2, -7)
Use the quadratic formula to solve the equation 6x² - 24x - 30 = 0.
Use the quadratic formula to solve the equation 6x² - 24x - 30 = 0.
5, -1
What are the exact solutions for the equation 2(x + 4)² - 48 = 0?
What are the exact solutions for the equation 2(x + 4)² - 48 = 0?
Signup and view all the answers
Simplify the expression (2 + 5i)/(5 + 4i).
Simplify the expression (2 + 5i)/(5 + 4i).
Signup and view all the answers
What is the solution set for the quadratic inequality x² - 2x ≥ 24?
What is the solution set for the quadratic inequality x² - 2x ≥ 24?
Signup and view all the answers
Write the quadratic equation in vertex form for y = x² + 14x + 4.
Write the quadratic equation in vertex form for y = x² + 14x + 4.
Signup and view all the answers
Which inequality is best represented by the graph?
Which inequality is best represented by the graph?
Signup and view all the answers
Use factoring and the Zero-Product Property to find the zeros of the quadratic function f(x) = x² - 7x - 8.
Use factoring and the Zero-Product Property to find the zeros of the quadratic function f(x) = x² - 7x - 8.
Signup and view all the answers
F(x) = -1 + 9x² + 7x is a quadratic function.
F(x) = -1 + 9x² + 7x is a quadratic function.
Signup and view all the answers
Find the value of x to the nearest hundredth.
Find the value of x to the nearest hundredth.
Signup and view all the answers
Solve the quadratic inequality x² - 4x + 3 < 0 and graph the solution on a number line.
Solve the quadratic inequality x² - 4x + 3 < 0 and graph the solution on a number line.
Signup and view all the answers
Factor the quadratic expression x² - 11x + 30.
Factor the quadratic expression x² - 11x + 30.
Signup and view all the answers
What is the quadratic equation that best fits the baseball's trajectory described?
What is the quadratic equation that best fits the baseball's trajectory described?
Signup and view all the answers
Write the function in the form f(x) = ax² + bx + c and identify a, b, and c for f(x) = (5x - 4)(5x + 3).
Write the function in the form f(x) = ax² + bx + c and identify a, b, and c for f(x) = (5x - 4)(5x + 3).
Signup and view all the answers
Find a quadratic equation that models the turnstile data. What is the equation?
Find a quadratic equation that models the turnstile data. What is the equation?
Signup and view all the answers
Solve the equation by completing the square for x² - 2x - 24 = 0. What are the solutions?
Solve the equation by completing the square for x² - 2x - 24 = 0. What are the solutions?
Signup and view all the answers
What equation represents the value remaining in Hans's account after x visits?
What equation represents the value remaining in Hans's account after x visits?
Signup and view all the answers
What is the value remaining in Hans's account after 11 visits?
What is the value remaining in Hans's account after 11 visits?
Signup and view all the answers
Solve the proportion (x−8)/8 = 4/5.
Solve the proportion (x−8)/8 = 4/5.
Signup and view all the answers
Solve for x in the equation x - 7 = 1.
Solve for x in the equation x - 7 = 1.
Signup and view all the answers
Solve for b in the formula B = 7a²b.
Solve for b in the formula B = 7a²b.
Signup and view all the answers
Which equation best describes the graph?
Which equation best describes the graph?
Signup and view all the answers
Solve for x in the equation 3x - 3 = x - 9.
Solve for x in the equation 3x - 3 = x - 9.
Signup and view all the answers
What inequality has the solution shown in the graph?
What inequality has the solution shown in the graph?
Signup and view all the answers
What inequality describes the graph?
What inequality describes the graph?
Signup and view all the answers
What is the equation of the line of best fit for the given data?
What is the equation of the line of best fit for the given data?
Signup and view all the answers
Write a proportion that models the statement 10 is to 5 as 16 is to 8.
Write a proportion that models the statement 10 is to 5 as 16 is to 8.
Signup and view all the answers
Write an equation in slope-intercept form for a line that passes through points (4,-7) and (-2,-10).
Write an equation in slope-intercept form for a line that passes through points (4,-7) and (-2,-10).
Signup and view all the answers
Which is the equation of the line with slope -4 and y-intercept 5?
Which is the equation of the line with slope -4 and y-intercept 5?
Signup and view all the answers
Identify the inequality which results in an acceptable number of golf balls in each bag.
Identify the inequality which results in an acceptable number of golf balls in each bag.
Signup and view all the answers
What will the population of Abbiville be in 2014 if growth remains constant?
What will the population of Abbiville be in 2014 if growth remains constant?
Signup and view all the answers
Match the following expressions with their evaluations:
Match the following expressions with their evaluations:
Signup and view all the answers
If x = 78 when y = 130, find x when y = 190.
If x = 78 when y = 130, find x when y = 190.
Signup and view all the answers
Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
Signup and view all the answers
What is the approximate average cost per T-shirt when the company makes 10 shirts?
What is the approximate average cost per T-shirt when the company makes 10 shirts?
Signup and view all the answers
What is the reduced row-echelon form of the given system of equations?
What is the reduced row-echelon form of the given system of equations?
Signup and view all the answers
Find the maximum and minimum values of C = 3x + 5y given the constraints.
Find the maximum and minimum values of C = 3x + 5y given the constraints.
Signup and view all the answers
Study Notes
Algebra Concepts and Solutions
- For athletes, an account balance can be modeled as V = 205 - 10x, where x represents visits. After 11 visits, the balance is $95.
- Solved proportion: (x−8)/8 = 4/5 results in x = 14 or 72/5.
- Simple equation solution: x - 7 = 1 leads to x = 8.
- Rearranged equation to isolate b: B = 7a²b becomes b = B/(7a²).
- Linear equation from a graph is represented as y = -2x + 5.
- Solve 3x - 3 = x - 9 to find x = -3.
- The inequality m + 7 ≥ 9 follows from a specific graph solution.
- For x ≤ -5, this describes another graph visually represented.
- The line of best fit for given data yields y = 0.63x + 1.90.
- Proportion of 10 to 5 as 16 to 8 is written as 10/5 = 16/8.
- Using points (4,-7) and (-2,-10), the slope-intercept form is y = 1/2x - 9.
- For points (-6, 5) and (-9, 0), the line is y = 5x/3 + 15.
- The equation for a parallel line through (-5, 4) to y = -4x + 1 is y = -4x - 16.
- Acceptable ranges for golf balls are modeled by the inequality: 365 ≥ 375.
- Estimating Abbiville's population using given data results in 364 inhabitants in 2014.
- The equation for a line with slope -4 and y-intercept 5 can be expressed as 4x + y - 5 = 0 or y = -4x + 5.
- Data from restaurant requests illustrates a positive linear relationship.
- The equation for a perpendicular line passing through (-1, -5) to -8x - 4y = 3 becomes y = -2x - 7.
- An equation with slope -4 through point (3, -6) is y = -4x + 6.
- Direct variation from y = 130 when x = 78 finds x = 114 when y = 190, with a constant k = 0.6.
- The transformation y = f(x-3) involves shifting the graph 3 units to the right.
- The evaluation of (2/5)⁵ results in 32/3125.
- The Commutative Property of Addition is exemplified: 5 + 2 = 2 + 5.
- Evaluating 64 × 4² - 2 × 2² gives 1016.
- Cost function for decorating stuffed animals: C(x) = 5.25x + 44.
- Total cost for 25 decorated animals comes to $175.25. A budget of $227.75 allows decorating 35 animals.
- Function composition: (g o f)(x) = x² - 12x + 34, (f o g)(x) = -x² + 8 for f(x) = 6 - x, g(x) = x² - 2.
- Evaluating (3/8)⁻² gives 64/9.
- From y = 15(x + 2) + 4, the graph translates up 4 units and stretches vertically by a factor of 5.
- Simplification of expression results in -u⁶⁰ from ((-u)³(-u⁹)⁸)/(u⁵)³.
- Squaring (-3/8) produces 9/64.
- The transformation of f(x) = x² by vertical stretching and translations yields g(x) = 3(x - 4)² - 8.
- Average cost function A(x) = (3.75x + 200)/x calculates to $23.75 for 10 shirts.
- Inverses of functions: for y = 7x², the inverse y = ±√(x/7) is not a function.
- A model for customer flow at The Burger Barn estimates 864,000 customers in 1999 from C(t) = t² + 38t + 600.
- Multiplication of functions generates (f × g)(x) = x³ - 3x² - 9x + 27 for f(x) = 9 - x², g(x) = 3 - x.
- Solve the quadratic equation 4x² = 8, yielding solutions ±√2 and approximate values ±1.41.
- Factoring the quadratic equation 2x² + 5x - 3 = 0 provides roots of 1/2 and -3.
- Quadratic fitting results in y = 0.09x² - 0.98x + 4.99 for points (1, 4.1), (5, 2.3), (11, 4.9).
- Solve quadratic inequalities graphically: x² - 2x ≥ 24 yields x ≤ -4 or x ≥ 6.
- Vertex form conversion for y = x² + 14x + 4 results in y = (x + 7)² - 45.
- Simplification of the expression (2 + 5i)/(5 + 4i) gives (30/41) + (17/41)i.
- Identifying solutions for quadratic inequality x² - 4x + 3 < 0 results in 1 < x < 3.
- Zero-finding of the function f(x) = x² - 7x - 8 yields roots 8 and -1.
- Analysis of matrices and equations provides solutions through various methods like elimination and substitution.### Quadratic Equations and Functions
- A quadratic equation takes the form y = ax² + bx + c where a, b, and c are constants.
- Example quadratic equation for baseball's trajectory: y = -0.0383x² + 2.143x + 50.
Identifying Quadratic Functions
- Given expression: f(x) = (5x - 4)(5x + 3) can be written in the standard form.
- Expansion results in: f(x) = 25x² - 5x - 12.
- Coefficients identified: a = 25, b = -5, c = -12.
Modeling Data with Quadratic Equations
- Turnstile data at State Fair can be modeled with the quadratic equation: y = 0.083x² - 0.197x + 7.62.
- Data analysis utilizes graphing calculators to fit quadratic models:
- Enter x values into list L1 and y values into list L2.
- Use Quadratic Regression function to derive the equation.
Solving Quadratic Equations by Completing the Square
- Example equation: x² - 2x - 24 = 0.
- Completing the square provides exact solutions: x = -4, x = 6.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.