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Questions and Answers
What does the vertex of a parabola represent in the context of quadratic equations?
What does the vertex of a parabola represent in the context of quadratic equations?
How can the x-intercepts of a quadratic function be determined using its graph?
How can the x-intercepts of a quadratic function be determined using its graph?
What does it mean if a quadratic graph opens upwards?
What does it mean if a quadratic graph opens upwards?
In the quadratic equation $x^2 + 6x + 9$, what does the term $9$ represent in the context of the graph?
In the quadratic equation $x^2 + 6x + 9$, what does the term $9$ represent in the context of the graph?
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What information does the axis of symmetry provide in a quadratic function?
What information does the axis of symmetry provide in a quadratic function?
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Study Notes
Envision Algebra 1 Revision
- Chapters covered: L1, L2, L3, L4, and L5
Solving Quadratic Equations Using Graphs and Tables (Ch9L1)
- Graphs show solutions to quadratic equations
- X-intercepts represent solutions
- A graph touching the x-axis indicates one solution
- A graph not crossing the x-axis indicates no real solutions
Solving Quadratic Equations by Graphing (Ch9L2)
- Factoring helps solve quadratic equations
- Factoring example: x² + 2x + 1 = 0 factors to (x + 1)(x + 1) = 0, giving a single solution x = -1
- Another example: x² - 5x - 14 = 0 factors to (x - 7)(x + 2) = 0, giving solutions x = 7 and x = -2
- Further example: x² + 7x = 0 factors to x(x + 7) = 0, giving solutions x = 0 and x = -7
- Final example: 2x² - 5x + 2 = 0 factors to (2x - 1)(x - 2) = 0, giving solutions x = 1/2 and x = 2
Example 4 Using Factored Form to Graph a Quadratic Function
- Given the function f(x) = x² - 2x - 8
- Factor it as (x + 2)(x - 4).
- x-intercepts are -2 and 4.
- Average the x-intercepts to find the x-coordinate of the vertex, which is 1.
- Substitute the value of x = 1 into the function to find the y-coordinate of the vertex, which is -9.
- Vertex is (1, -9).
- Plot the vertex and x-intercepts to graph the quadratic.
Write the factored form for the quadratic function (Ch9L2)
- (x + 4)(x - 3) = 0 (From the graph, the x-intercepts are -4 and 3)
Rewriting Radical Expressions (Ch9L3)
- Simplifying expressions involving square roots
- Examples:
- √12x • √3x = 6x
- 2x⁹ • √26x⁶ = 2x⁷ √13x
- √27m • √6m²⁰ = 9m¹⁰√2m
- √2x³ • √25x²y = 5x² √2xy
Solving Quadratic Equations Using Square Roots (Ch9L4)
-
Solving equations of the form x² = a
- x² = 256 → x = ±16
- x² = 144 → x = ±12
- x² = -20 → no solution
- x² = -27 → no solution
-
Solving example problems
- x² + 65 = 90 → x = ±5
- x² - 65 = 90 → x = ±√155
- 3x² + 8 = 56 → x = ± √16 /3 = ± 4/√3 = ±4/√3
- 8x² - 40 = -470 → x = ± √-470 /8 no real solutions
Completing the Square (Ch9L5)
- Completing the square to solve quadratic equations
- Example problem (example from Ch9L5): x² - 14x + 16 = 0
- Steps to solve, as in the example, by completing the square and finding solutions: x = 7 ± √33
- Example: m² + 16m = -59, solution -8 ± √5
- Example: x² - 2x - 35 = 0, solutions -5 and 7
- Example 4 (finding vertex form from example page 14):
- Given equation: y = x² - 8x + 11.
- The vertex form: y = (x - 4)² - 5
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Description
This quiz covers key concepts from Algebra 1 related to solving quadratic equations, specifically through graphs and factoring methods. Review the properties of x-intercepts and practice factoring equations to find solutions. Strengthen your understanding of quadratic functions as you prepare for exams.
