2.2 Quadratics (Quadratic Functions)

1. The Simplest Quadratic: $y = x^2$

The graph of a quadratic is a curve known as a parabola. For the simplest quadratic $y = x^2$:

  • The domain evaluates to $x \in \mathbb{R}$ (all real numbers).
  • The range evaluates to $y \ge 0$ (or $[0, +\infty)$).
  • The function $y = -x^2$ is an inverted reflection across the x-axis.

2. The Quadratic Function $y = ax^2 + bx + c$

The basic characteristics of the generalized quadratic graph are as follows:

1) Concavity: Determined by $a \ne 0$.
If $a > 0$, the graph is concave up (U-shaped). If $a < 0$, it is concave down.
2) Discriminant ($\Delta$): $\Delta = b^2 - 4ac$ determines the root count.
$\Delta > 0$: 2 real roots. $\Delta = 0$: 1 real root. $\Delta < 0$: No real roots.
3) Roots (x-intercepts): $x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}$ (exists only if $\Delta \ge 0$).
4) y-intercept: Set $x = 0$ to obtain $y = c$.
5) Axis of Symmetry: Given by the equation $x = -\frac{b}{2a}$. This also represents the x-coordinate of the vertex. Alternatively, if roots $x_1$ and $x_2$ are known, the vertex lies at the midpoint $x = \frac{x_1 + x_2}{2}$.

EXAMPLE 1

Analyze the function $y = 2x^2 - 12x + 10$.

$a = 2 > 0$, so the graph is concave up.
$\Delta = (-12)^2 - 4(2)(10) = 64 > 0$, confirming two roots.
Roots: $x_{1,2} = \frac{12 \pm 8}{4} \Rightarrow x = 1$ and $x = 5$.
y-intercept: $y = 10$.
Axis of symmetry: $x = -\frac{-12}{4} = 3$. At $x=3$, $y = 2(9) - 36 + 10 = -8$. The vertex is $V(3, -8)$.

3. Quadratic Inequalities

To solve inequalities such as $ax^2 + bx + c > 0$, evaluate the roots and the concavity to graphically identify the correct intervals.

EXAMPLE 2

Let $f(x) = 2x^2 - 4x + k$. Determine the values of $k$ based on specific conditions.

The discriminant is $\Delta = 16 - 8k$.
(a) Exactly one root: $\Delta = 0 \Rightarrow 16 - 8k = 0 \Rightarrow k = 2$.
(b) Exactly two roots: $\Delta > 0 \Rightarrow 16 - 8k > 0 \Rightarrow k < 2$.
(c) No real roots: $\Delta < 0 \Rightarrow 16 - 8k < 0 \Rightarrow k > 2$.
(d) Has real roots (one or two): $\Delta \ge 0 \Rightarrow k \le 2$.
(e) $f(x) > 0$ for any $x$: Requires the parabola to float above the x-axis (no roots, concave up). $\Delta < 0 \Rightarrow k > 2$.
(f) $f(x) \ge 0$ for any $x$: Can touch the axis once or not at all. $\Delta \le 0 \Rightarrow k \ge 2$.

4. Forms of a Quadratic Function

  • Traditional form: $y = ax^2 + bx + c$
  • Factorization form: $y = a(x - r_1)(x - r_2)$ (where $r_1, r_2$ are roots)
  • Vertex form: $y = a(x - h)^2 + k$ (where $(h,k)$ is the vertex)

Notice: A quadratic can be manually converted to vertex form by "completing the square," e.g., $2x^2 - 12x + 10 = 2(x^2 - 6x + 9 - 9) + 10 = 2(x-3)^2 - 8$.

EXAMPLE 3 & 4 & 5

Example 3: Convert $y = 2x^2 - 12x + 10$.
Roots are 1 and 5: Factored form is $y = 2(x-1)(x-5)$.
Vertex is $(3, -8)$: Vertex form is $y = 2(x-3)^2 - 8$.
Example 4: Let $y = -3x^2 - 15x + 42$.
Roots are $-7$ and $2$: $y = -3(x+7)(x-2)$.
Vertex is $(-2.5, 60.75)$: $y = -3(x+2.5)^2 + 60.75$.
Example 5: Consider $f(x) = 3x^2 + 12x$.
Factored: $3x(x+4)$. Roots: $x=0, x=-4$.
Vertex x-coordinate: $-12 / 6 = -2$. Vertex: $(-2, -12)$.
Vertex form: $3(x+2)^2 - 12$.

5. Vieta Formulas

Given the quadratic $y = ax^2 + bx + c$ with real roots $r_1, r_2$:

Sum of roots (S): $S = r_1 + r_2 = -\frac{b}{a}$
Product of roots (P): $P = r_1 r_2 = \frac{c}{a}$

Conversely, knowing the sum and product constructs the base quadratic: $y = x^2 - Sx + P$.

EXAMPLE 6

For the function $y = 2x^2 - 12x + 10$, the roots are 1 and 5.

Sum: $S = -\frac{-12}{2} = 6$.
Product: $P = \frac{10}{2} = 5$.
Reconstructing: $x^2 - 6x + 5 = 0$, which scales up to the original $2x^2 - 12x + 10 = 0$.