Quadratic Functions and Equations

Quadratic Equations

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

• $a$ is the quadratic coefficient.

• $b$ is the linear coefficient.

• $c$ is the constant term.

Sum and Product of Roots

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

• Sum of roots: $-\frac{b}{a}$

• Product of roots: $\frac{c}{a}$

Example Problems

1. 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 = $-\frac{-16}{2} = \frac{16}{2} = 8$

2. 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 = $-\frac{-16}{1} = 16$

3. 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: $-\frac{0}{1} = 0$

4. 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 = $\frac{-35}{1} = -35$

5. 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 = $\frac{6}{1} = 6$

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 = $\frac{-10}{-2} = -3$

Factors and Roots/Solutions

• Factors: $(x - 3)$ and $(x + 2)$

• Roots/Solutions: $x = 3$ and $x = -2$

• X-intercepts: where $y = 0$

Problems Involving Factors and Roots

1. 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:
       $(-1)^3 - 5(-1)^2 + k(-1) + 2 = 0$
       $-1 - 5 - k + 2 = 0$
       $-4 - k = 0$
       $k = -4$

2. 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:
       $(1)^3 - 3(1)^2 + 2(1) - 4b = 0$
       $1 - 3 + 2 - 4b = 0$
       $0 - 4b = 0$
       $b = 0$

Constant Determination

1. 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:
       $(2)^2 - 11(2) + k = 0$
       $4 - 22 + k = 0$
       $-18 + k = 0$
       $k = 18$

2. 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:
       $0 = (3)^2 - 3a - 3$
       $9 - 3a - 3 = 0$
       $6 - 3a = 0$
       $-3a = -6$
       $a = 2$

Quadratic Functions

General Forms

1. Y-intercept Form:

   • $f(x) = ax^2 + bx + c$

   • Y-intercept: $(0, c)$

2. Factored Form (X-intercept Form):

   • $f(x) = a(x - x<em>1)(x - x</em>2)$

   • X-intercepts: $(x<em>1, 0)$, $(x</em>2, 0)$

3. Vertex Form:

   • $f(x) = a(x - h)^2 + k$

   • Vertex: $(h, k)$

Key Concepts

• 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.

Vertex Calculation

• The x-coordinate (abscissa) of the vertex is given by: $h = -\frac{b}{2a}$

Example Problems

1. For the quadratic function $f(x) = 3x^2 + 6x - 9$

   • The y-intercept is $(0, -9)$

   • The x-coordinate of the vertex is: $x = -\frac{6}{2(3)} = -1$

2. Problem: What is the abscissa of the vertex of the function $f(x) = 3x^2 - 18x + 4$?

   • Solution:

     • Using the formula $h = -\frac{b}{2a}$:
       $h = -\frac{-18}{2(3)} = \frac{18}{6} = 3$

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 $h = -\frac{b}{2a}$:
       $h = -\frac{-2}{2(5)} = \frac{2}{10} = \frac{1}{5}$

4. 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:

     • $-2.5 = -\frac{a}{2(2)}$

     • $-2.5 = -\frac{a}{4}$

     • $a = 10$

5. 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 $h + k$:

       • $h + k = 2 + 2 = 4$

6. If $f(x) = ax^2 + bx + c$ is concave downward, which of the following must be true?

   • (A) a < 0

7. For the equation $y = 2x^2 + 10x + 12$. What is the value of k if the graph crosses the y-axis at the point $(0, k)$?

• By substituting $x = 0$:

  • $y = 2(0)^2 + 10(0) + 12$

  • $y = 12$

Identifying Equations from Graphs

1. X-intercept Form:

   • Given x-intercepts at $1/5$ and $-3/2$, the equation could be: $y = (5x - 1)(2x + 3)$

2. 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:

       • $y = 7(x - 1)(x - 3)$

Determining Equation from Graph

1. 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

  • $y = (x - 1)(x - 5)$

1. October 2020 EST Section 3

• 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 $k$?

  • $x1=-2$

  • $x2=6$

  • $h = (6-2)/2 = 2$

1. March 2021 EST Section 4

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

1. Official test 7 sec 3

• 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 ?

  • $h = -3 - 1 = -2$

1. Panda test 6 sec 4

• 30 y = (2x-1)(2x-11)

1. December 2020 EST Section 3

• 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)

Vertex Form

• $f(x) = a(x - h)^2 + k$

• Vertex = $(h, k)$

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

Ethical implications

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