Review of Algebraic Expressions and Equations
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Questions and Answers

Write two equivalent expressions for 15x - 10x + 5x.

8x + 7x

Write two equivalent expressions for 12x - 6x + 4 * 2x * 5.

4x³ * 3x⁶

Create a monomial and a trinomial. Then, multiply them together to get an equivalent sum.

3x(x² + 2x - 5) = 3x³ + 6x² - 15x

Create a binomial and a binomial. Then, multiply them together to get an equivalent sum. Show your work.

<p>(2x + 3)(x - 4) = 2x² - 5x - 12</p> Signup and view all the answers

Create 3 binomials. Then, multiply them together to get an equivalent sum. Show your work.

<p>(x + 2)(x + 3)(x + 4) = x³ + 19x² + 26x + 24</p> Signup and view all the answers

Which of the following expressions are equivalent to 3 + x - (2x² + 5x - 1)? For each option, state if it's equivalent or not, and explain why.

<p>-2(x² + 2x - 2)</p> Signup and view all the answers

Write an equivalent sum for 4x²(7x - 3) using multiplication.

<p>28x³ - 12x²</p> Signup and view all the answers

Write an equivalent sum for 3x - 2 - (5x² + 4x - 8) using multiplication.

<p>-5x² - x + 6</p> Signup and view all the answers

Write an equivalent sum for -3xy²(4xy⁷ + 6x²y³) using multiplication.

<p>-12x²y⁹ - 18x³y⁶</p> Signup and view all the answers

Write an equivalent sum for (2x - 7)(x + 6) using multiplication.

<p>2x² + 5x - 42</p> Signup and view all the answers

Write an equivalent sum for 5x⁴(3x² - 4x + 1) using multiplication.

<p>15x⁶ - 20x⁵ + 5x⁴</p> Signup and view all the answers

Write an equivalent sum for (x + 3)(2x² - 7x + 9) using multiplication.

<p>2x³ - x² - 12x + 27</p> Signup and view all the answers

Write an equivalent product for x² + x - 42 by factoring.

<p>(x + 7)(x - 6)</p> Signup and view all the answers

Write an equivalent product for 16 - 25x² by factoring.

<p>(4 + 5x)(4 - 5x)</p> Signup and view all the answers

Write an equivalent product for -4x⁵ - 14x⁴ by factoring.

<p>-2x⁴(2x + 7)</p> Signup and view all the answers

Write an equivalent product for 3x³ - 75x by factoring.

<p>3x(x - 5)(x + 5)</p> Signup and view all the answers

Write an equivalent product for 2x³ - 5x² - 18x + 45 by factoring.

<p>(x - 3)(x + 3)(2x - 5)</p> Signup and view all the answers

Write an equivalent product for 81 - 16x⁴ by factoring.

<p>(3 - 2x)(3 + 2x)(9 + 4x²)</p> Signup and view all the answers

Write an equivalent product for 4x⁴ - 16x³ + 20x² by factoring.

<p>4x²(x² - 4x + 5)</p> Signup and view all the answers

Write an equivalent product for 6x³ + 3x² - 18x by factoring.

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

Create a sum that can be factored into a product using the GCF method. Then, factor it.

<p>2x² - 8x = 2x(x - 4)</p> Signup and view all the answers

Create a sum that can be factored into a product using the Difference of Squares method. Then, factor it.

<p>x² - 9 = (x + 3)(x - 3)</p> Signup and view all the answers

Create a sum that can be factored into a product using the Trinomial Shortcut method. Then, factor it.

<p>x² - 2x - 8 = (x - 4)(x + 2)</p> Signup and view all the answers

Create a sum that must first be factored using GCF, then can be factored further using either Difference of Squares or the Trinomials Shortcut methods. Then, factor it.

<p>3x³ - 12x = 3x(x - 2)(x + 2)</p> Signup and view all the answers

Which of the following expressions are quadratic functions? Explain why or why not.

<p>g(x) = 4x² + 11x - 3</p> Signup and view all the answers

How many possible x-intercepts can a quadratic equation have? Sketch graphs to show this.

<p>None</p> Signup and view all the answers

Which part of the Quadratic Formula tells you how many x-intercepts your quadratic function will have? Explain how you can use this part to determine the number of x-intercepts.

<p>The discriminant (b² - 4ac)</p> Signup and view all the answers

Find the x-intercepts of the function f(x) = 6x² + 12x by factoring and using the Zero Product Property. Show all work.

<p>(0, 0) and (-2, 0)</p> Signup and view all the answers

Find the x-intercepts of the function h(x) = x² - 10x + 25 by factoring and using the Zero Product Property. Show all work.

<p>(5, 0)</p> Signup and view all the answers

Find the x-intercepts of the function g(x) = 3x² + 10x - 8 by factoring and using the Zero Product Property. Show all work.

<p>(2/3, 0) and (-4, 0)</p> Signup and view all the answers

Find the x-intercepts of the function y = 4x² - 9 by factoring and using the Zero Product Property. Show all work.

<p>(3/2, 0) and (-3/2, 0)</p> Signup and view all the answers

Find the y-intercept of the function f(x) = x² - 2x - 8. Show all work.

<p>(0, -8)</p> Signup and view all the answers

Find the y-intercept of the function g(x) = 3x² - 9x. Show all work.

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

Find the y-intercept of the function y = 5x² + 4x - 7. Show all work.

<p>(0, -7)</p> Signup and view all the answers

Create a quadratic equation that has a y-intercept of (0, 7). Show your work to prove that it does.

<p>y = x² + 7</p> Signup and view all the answers

Find the x-intercepts of the function f(x) = 4x² + 12x + 9 using the Quadratic Formula. Show all work.

<p>(−1.5, 0) and (−1.5, 0)</p> Signup and view all the answers

Find the x-intercepts of the function 9 = 2x² + 3x using the Quadratic Formula. Show all work.

<p>(1.5, 0) and (−3, 0)</p> Signup and view all the answers

Create a quadratic equation in standard form (y = ax² + bx + c) that has no x-intercepts. The c-value must be negative. Show work to prove that it does not have x-intercepts.

<p>y = −2x² + x − 9</p> Signup and view all the answers

Find the vertex of the function y = (x + 4)(x + 2) using the most appropriate method. Show all work.

<p>(-3, -1)</p> Signup and view all the answers

Find the vertex of the function f(x) = 2x² + 5x + 4 using the most appropriate method. Show all work.

<p>(-1.25, 0.875)</p> Signup and view all the answers

Is x = -4 the line of symmetry for the parabola f(x) = (x - 5)(x - 3)? Show why or why not.

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

Simplify the following radicals: √54

<p>3√6</p> Signup and view all the answers

Find the x-intercepts of the function f(x) = 2x² - 4x - 8 using the Quadratic Formula. Write your x-intercepts in exact form (simplify any radicals and fractions) and in approximate form. Show all work.

<p>Exact: (1 ± √5, 0), Approximate: (-1.24, 0) and (3.24, 0)</p> Signup and view all the answers

Find the x-intercepts of the function y = 4x² + 6x - 8 using the Quadratic Formula. Write your x-intercepts in exact form (simplify any radicals and fractions) and in approximate form. Show all work.

<p>Exact: (-3/2 ± √41/4, 0) Approximate: (-2.35, 0) and (0.85, 0)</p> Signup and view all the answers

Find the x-intercepts of the function 2x² + 12x + 9 using the Quadratic Formula. Write your x-intercepts in exact form (simplify any radicals and fractions) and in approximate form. Show all work.

<p>Exact: (-3 ± √72/4, 0) Approximate: (-5.12, 0) and (-1.88, 0)</p> Signup and view all the answers

Graph the parabola f(x) = x² - 2x - 3 by finding its x-intercepts, vertex, y-intercept, axis of symmetry, and a-value. Choose your method for finding the vertex. Show all work.

<p>x-intercepts: (3, 0) and (-1, 0), vertex: (1, -4), y-intercept: (0, -3), axis of symmetry: x = 1, a-value: 1</p> Signup and view all the answers

Graph the parabola g(x) = 2x² - 10x by finding its x-intercepts, vertex, y-intercept, axis of symmetry, and a-value. Choose your method for finding the vertex. Show all work.

<p>x-intercepts: (0, 0) and (5, 0), vertex: (2.5, -12.5), y-intercept: (0, 0), axis of symmetry: x = 2.5, a-value: 2</p> Signup and view all the answers

Write the equation in vertex form of a parabola that has a vertex at (3, −1).

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

Take the equation y = (x - 3)² - 1 and make it open down.

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

Take the equation y = (x - 3)² - 1 and make it vertically stretched.

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

Write the equation in vertex form of a parabola that opens up, is vertically compressed, and has a vertex at (-2, 5).

<p>y = (1/2)(x + 2)² + 5</p> Signup and view all the answers

What is the vertex of the function y = -x² - 5? State whether it opens up or down, is vertically stretched or compressed, if it shifts left/right (how much), and if it shifts up/down (how much).

<p>Vertex: (0, -5), opens down, vertically compressed, no shift left/right, vertical translation down 5</p> Signup and view all the answers

Complete the square to put the function f(x) = x² - 8x + 2 into vertex form. Then, state the vertex.

<p>f(x) = (x - 4)² - 14, vertex: (4, -14)</p> Signup and view all the answers

Create your own quadratic equation in standard form, y = ax^2 + bx + c, where the a-value is 1. Then, complete the square to put it in vertex form and state the vertex.

<p>y = x² + 6x + 10, vertex: (-3, 1)</p> Signup and view all the answers

Complete the square to put the function y = 2x² + 8x - 5 into vertex form. Then, state the vertex.

<p>y = 2(x + 2)² - 13, vertex: (-2, -13)</p> Signup and view all the answers

Complete the square to put the function f(x) = -5x^2 + 30x - 12 into vertex form. Then, state the vertex.

<p>f(x) = -5(x - 3)² + 33, vertex: (3, 33)</p> Signup and view all the answers

What information about the parabola can you get immediately (without showing work) from a quadratic equation in standard form, y = ax² + bx + c?

<p>y-intercept</p> Signup and view all the answers

What information about the parabola can you get immediately (without showing work) from a quadratic equation in vertex form, y = a(x - h)² + k?

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

Create your own quadratic equations in vertex form, y = a(x - h)² + k, that have the given amount of x-intercepts. Show work to prove that they have the correct amount of x-intercepts.

<p>a. y = (x - 2)² - 4, b. y = (x + 3)² , c. y = (x - 1)² + 5</p> Signup and view all the answers

Find the vertex, x-intercepts, y-intercept, and make a graph of the function y = (x + 2)² + 9. Be sure to plot the a-value on the graph. Show all work.

<p>Vertex: (-2, 9), x-intercepts: (-5, 0) and (1, 0), y-intercept: (0, 5), a-value: 1 (plotted on the graph).</p> Signup and view all the answers

Find the vertex, x-intercepts, y-intercept, and make a graph of the function f(x) = 2(x - 2)² . Be sure to plot the a-value on the graph. Show all work.

<p>Vertex: (2, 0), x-intercepts: (2, 0), y-intercept: (0, 8), a-value: 2 (plotted on the graph).</p> Signup and view all the answers

Find the vertex, x-intercepts, y-intercept, and make a graph of the function f(x) = -(1/2)(x - 3)² + 2 . Be sure to plot the a-value on the graph. Show all work.

<p>Vertex: (3, 2), x-intercepts: (5, 0) and (1, 0), y-intercept: (0, -2.5), a-value: -1/2 (plotted on the graph).</p> Signup and view all the answers

Find the vertex, x-intercepts, y-intercept, and make a graph of the function f(x) = 2(x - 1)² + 4 . Be sure to plot the a-value on the graph.

<p>Vertex: (1, 4), x-intercepts: None, y-intercept: (0, 6), a-value: 2 (plotted on the graph).</p> Signup and view all the answers

Study Notes

Review of Equations and Expressions

  • Equivalent expressions are different ways of writing the same mathematical expression.
  • This includes simplifying expressions such as combining like terms in order to determine the equivalence of given sets of equations.
  • A monomial is a single term, a binomial consists if 2 terms and a trinomial consists of 3 terms.
  • Binomials can be applied across different algebraic terms (equations).
  • Factoring is the opposite of multiplication as demonstrated with different algebraic terms.
  • In some cases, equations/ expressions do not have equivalent forms, or in more complex expressions, are not factorable.
  • A quadratic expression is one that contains a squared term and other terms.
  • The discriminant (b² - 4ac) in the quadratic formula indicates how many x-intercepts there are.
  • A quadratic function has a maximum of two x-intercepts.
  • The x-intercepts (roots) of a quadratic equation are where the graph intersects the x-axis.
  • The y-intercept is where the graph intersects the y-axis

Quadratic Functions

  • Quadratic functions are mathematical expressions containing a squared term (x²)
  • Quadratics have a maximum of 2 x-intercepts
  • The discriminant is part of the quadratic formula that indicates the number of x-intercepts (roots) from a quadratic equation.
  • The vertex of a quadratic equation can be found using various methods such as factoring or using the vertex formula.

Vertex Form

  • Vertex form of a quadratic equation is written as y = a(x - h)² + k, where (h,k) is the vertex.
  • The vertex is a critical point that indicates the turning point (maximum or minimum) of the parabola.
  • Completing the square is a method for converting a quadratic equation from standard form to vertex form.

Transformations to Functions

  • Changing the value of a in a function can affect the steepness/ shallowness/ vertical stretch or compression
  • The shifts left or right (horizontal translations) are expressed by (x-h)
  • The shifts up or down (vertical translations) are expressed by (k)
  • Reflecting a graph across the x-axis is done mathematically by multiplying all y-values by -1. A general rule applies- changes to functions generally affect the shape or location of the graph.
  • The locator point is the center point of a transformations that has been made to a function.
  • The 'a' value is an important element that indicates any transformational changes that have been done to the given function.

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Description

Test your understanding of algebraic expressions, including monomials, binomials, and trinomials. This quiz covers factoring, equivalence, and the properties of quadratic expressions. Evaluate your knowledge of the quadratic formula and its discriminant.

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