Quadratic Functions and Equations

A quadratic equation is expressed in the form ax^2 + bx + c = 0, where:

For a quadratic equation ax^2 + bx + c = 0, the sum and product of the roots can be determined using the following formulas:

Problem: What is the sum of all values of m that satisfy 2m^2 - 16m + 8 = 0?
- Solution:
  - Here, a = 2 and b = -16.
  - Sum of roots = -\frac{-16}{2} = \frac{16}{2} = 8
Problem: What is the sum of the solutions to the equation x^2 - 14x + 40 = 2x + 1?
- Solution:
  - First, rewrite the equation in standard quadratic form: x^2 - 14x + 40 - 2x - 1 = 0 \Rightarrow x^2 - 16x + 39 = 0
  - Here, a = 1 and b = -16.
  - Sum of roots = -\frac{-16}{1} = 16
Problem: What is the sum of the roots of the equation x^3 - 3x + 52 = 0?
- Solution:
  - rewrite the equation as x^3 + 0x^2 - 3x + 52 = 0
  - Here the coefficient of x^2 is 0, so the sum of the roots is: -\frac{0}{1} = 0
Problem: What is the product of the solutions to the equation x(x - 2) = 35?
- Solution:
  - First, rewrite the equation in standard quadratic form: x^2 - 2x - 35 = 0
  - Here, a = 1 and c = -35.
  - Product of roots = \frac{-35}{1} = -35
Problem: What is the product of the two solutions of the equation x^2 - 5x + 8 = 2?
- Solution:
  - First, rewrite the equation in standard quadratic form: x^2 - 5x + 8 - 2 = 0 \Rightarrow x^2 - 5x + 6 = 0
    - Here, a = 1 and c = 6.
  - Product of roots = \frac{6}{1} = 6
Problem: What is the product of the solutions of 2x^2 - 4 = 4x^2 + 6?
- Solution:
  - Rewrite the equation: 2x^2 - 4x^2 - 4 - 6 = 0 \Rightarrow -2x^2 - 10 = 0
  - Here, a = -2 and c = -10.
  - Product of roots = \frac{-10}{-2} = -3

Problem: If (x + 1) is a factor of x^3 - 5x^2 + kx + 2, then what is the value of k?
- Solution:
  - Since (x + 1) is a factor, x = -1 is a root. Substitute x = -1 into the equation:
    (-1)^3 - 5(-1)^2 + k(-1) + 2 = 0
    -1 - 5 - k + 2 = 0
    -4 - k = 0
    k = -4
Problem: If both (x - 1) and (x - 2) are factors of x^3 - 3x^2 + 2x - 4b, then what must b be?
- Solution:
  - Since (x - 1) is a factor, x = 1 is a root. Substitute x = 1 into the equation:
    (1)^3 - 3(1)^2 + 2(1) - 4b = 0
    1 - 3 + 2 - 4b = 0
    0 - 4b = 0
    b = 0

Problem: Given the function P(x) = x^2 - 11x + k, where k is a constant. If 2 is a zero of the function, what is the value of k?
- Solution:
  - Since 2 is a zero, P(2) = 0. Substitute x = 2 into the equation:
    (2)^2 - 11(2) + k = 0
    4 - 22 + k = 0
    -18 + k = 0
    k = 18
Problem: The function h(x) = x^2 - ax - 3 has zeros at x = 3 and x = -1. What is the value of a?
- Solution:
  - Since 3 is a zero, h(3) = 0. Substitute x = 3 into the equation:
    0 = (3)^2 - 3a - 3
    9 - 3a - 3 = 0
    6 - 3a = 0
    -3a = -6
    a = 2

Y-intercept Form:
- f(x) = ax^2 + bx + c
- Y-intercept: (0, c)
Factored Form (X-intercept Form):
- f(x) = a(x - x1)(x - x2)
- X-intercepts: (x1, 0), (x2, 0)
Vertex Form:
- f(x) = a(x - h)^2 + k
- Vertex: (h, k)

Graphs: Parabolas that open upwards or downwards.
Vertex: The minimum or maximum point of the parabola.
Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = h.
Range: The set of all possible output values (y-values) of the function.

For the quadratic function f(x) = 3x^2 + 6x - 9
- The y-intercept is (0, -9)
- The x-coordinate of the vertex is: x = -\frac{6}{2(3)} = -1
Problem: What is the abscissa of the vertex of the function f(x) = 3x^2 - 18x + 4?
- Solution:
  - Using the formula h = -\frac{b}{2a}:
    h = -\frac{-18}{2(3)} = \frac{18}{6} = 3
Problem: For the function f(x) = 5x^2 - 2x - 8, for what value of x does f(x) obtain its minimum value?
- Solution:
  - Using the formula h = -\frac{b}{2a}:
    h = -\frac{-2}{2(5)} = \frac{2}{10} = \frac{1}{5}
Problem: Given f(x) = 2x^2 + ax - 1. What is the value of a if the axis of symmetry of the graph of f is equal to -2.5?
- Solution: The axis of symmetry is x = -\frac{b}{2a}, so:
  - -2.5 = -\frac{a}{2(2)}
  - -2.5 = -\frac{a}{4}
  - a = 10
Problem: The graph of the function f(x) = -2x^2 + 8x - 6 is a parabola. If point S(h, k) is the vertex of the parabola, what is the value of h + k?
- Solution:
  - First, find h (the x-coordinate of the vertex):
    - h = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2
  - Next, find k (the y-coordinate of the vertex) by substituting h into the function:
    - k = f(2) = -2(2)^2 + 8(2) - 6 = -8 + 16 - 6 = 2
  - Finally, find h + k:
    - h + k = 2 + 2 = 4
If f(x) = ax^2 + bx + c is concave downward, which of the following must be true?
- (A) a < 0
For the equation y = 2x^2 + 10x + 12. What is the value of k if the graph crosses the y-axis at the point (0, k)?

X-intercept Form:
- Given x-intercepts at 1/5 and -3/2, the equation could be: y = (5x - 1)(2x + 3)
Problem: The graph of the equation y = 7x^2 - 28x + 21 is a parabola in the xy-plane. In which of the following equivalent forms of the equation do the x-intercepts of the parabola appear as constants or coefficients?
- Solution:
  - The x-intercepts appear as constants or coefficients in the factored form:
    - y = 7(x - 1)(x - 3)

Problem: Which of the following is an equivalent form of the equation y = x^2 - 6x + 5 that displays the x-intercepts of the parabola as constants?

Solution:
- The Factored form or X-intercept form displays the x-intercepts
- y = (x - 1)(x - 5)

The graph of the function f(x) = 2(x + 2) (x - 6) is a parabola. If the line x=k is the axis of symmetry of the parabola, what is the value of k?
- x1=-2
- x2=6
- h = (6-2)/2 = 2

Consider the function f defined by f(x)=2(x-3)(x+2). What is the ordinate of the vertex of function f?

The function f is defined by f(x) = (x+3)(x+1). The graph of ƒ in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of ƒ ?
- h = -3 - 1 = -2

14.The graph of the function f in the xy-plane above is a parabola. Which of the following expressions defines f while showing the x-intercepts as constants or coefficients?
- f(x) = x(x+2)

Problem:
y=-\frac{1}{2}(x-3)^2+a

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