5.1 Quadratic Equations and Functions

Quadratic equations and functions center on that $x^2$ term, and they're the foundation for understanding parabolas. Mastering them gives you tools to solve problems involving projectile motion, optimization, area, and much more.

This section covers the key features of quadratic functions, multiple methods for solving quadratic equations, graphing through transformations, and how domain and range work for parabolas.

Key features of quadratic functions

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Characteristics of quadratic functions

A quadratic function is a polynomial function of degree 2, meaning the highest exponent on the variable is 2. The standard form is:

$f(x) = ax^2 + bx + c$

where $a$, $b$, and $c$ are real numbers and $a \neq 0$. (If $a$ were zero, the $x^2$ term would vanish and you'd just have a linear function.)

The graph of every quadratic function is a parabola, a symmetrical U-shaped curve.

Components of a parabola

• Vertex: the point where the parabola changes direction. It's either the minimum (when the parabola opens up) or the maximum (when it opens down). You find the x-coordinate of the vertex using:

$x = \frac{-b}{2a}$

Then plug that x-value back into $f(x)$ to get the y-coordinate.

• Axis of symmetry: the vertical line that passes through the vertex, splitting the parabola into two mirror-image halves. Its equation is also $x = \frac{-b}{2a}$.
• Direction of opening: determined by the sign of $a$.
  • If $a > 0$, the parabola opens upward (vertex is a minimum).
  • If $a < 0$, the parabola opens downward (vertex is a maximum).
• y-intercept: found by substituting $x = 0$ into the function, which gives you $f(0) = c$. So the y-intercept is always the point $(0, c)$.
• x-intercepts (also called zeros or roots): found by setting $f(x) = 0$ and solving for $x$. A parabola can have two, one, or zero real x-intercepts depending on the discriminant (more on that below).

Solving quadratic equations

A quadratic equation in standard form is $ax^2 + bx + c = 0$, where $a \neq 0$. There are three main methods for solving these.

Factoring method

Factoring works by rewriting the quadratic as a product of two linear expressions, then applying the zero-product property: if $AB = 0$, then $A = 0$ or $B = 0$.

Steps for factoring when $a = 1$ (leading coefficient is 1):

1. Write the equation in standard form: $x^2 + bx + c = 0$.
2. Find two numbers that multiply to $c$ and add to $b$.
3. Write the factored form: $(x + m)(x + n) = 0$, where $m$ and $n$ are those two numbers.
4. Set each factor equal to zero and solve.

When $a \neq 1$, you need two numbers that multiply to $ac$ and add to $b$. Use those to split the middle term, then factor by grouping.

Example: Solve $x^2 + 5x + 6 = 0$. You need two numbers that multiply to 6 and add to 5. That's 2 and 3. $(x + 2)(x + 3) = 0$ $x = -2$ or $x = -3$

Completing the square method

This method rewrites the equation so one side is a perfect square. It's especially useful for deriving vertex form.

1. Start with $ax^2 + bx + c = 0$. If $a \neq 1$, divide every term by $a$.
2. Move the constant term to the right side.
3. Take half the coefficient of $x$, square it, and add that value to both sides.
4. Factor the left side as a perfect square: $(x + p)^2 = q$.
5. Take the square root of both sides (don't forget $\pm$) and solve for $x$.

Example: Solve $x^2 + 6x + 2 = 0$.

1. Move the constant: $x^2 + 6x = -2$
2. Half of 6 is 3, and $3^2 = 9$. Add 9 to both sides: $x^2 + 6x + 9 = 7$
3. Factor: $(x + 3)^2 = 7$
4. Solve: $x + 3 = \pm\sqrt{7}$, so $x = -3 \pm \sqrt{7}$

Quadratic formula

The quadratic formula solves any quadratic equation:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the radical, $b^2 - 4ac$, is called the discriminant. It tells you the nature of the solutions before you even solve:

• Positive discriminant → two distinct real solutions
• Zero discriminant → one repeated real solution (the parabola touches the x-axis at exactly one point)
• Negative discriminant → no real solutions; instead you get two complex conjugate solutions (you'll work with these more in the complex numbers part of this unit)

Graphing quadratic functions

Transformations of quadratic functions

Every quadratic function can be understood as a transformation of the parent function $f(x) = x^2$. The most useful form for graphing transformations is vertex form:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex. Each parameter controls a specific transformation:

• Vertical shift: the $k$ value moves the graph up ($k > 0$) or down ($k < 0$).
• Horizontal shift: the $h$ value moves the graph right ($h > 0$) or left ($h < 0$). Watch the sign carefully: $(x - 3)^2$ shifts right 3, while $(x + 3)^2 = (x - (-3))^2$ shifts left 3.

Reflections and dilations

• Reflection across the x-axis happens when $a$ is negative. This flips the parabola so it opens in the opposite direction.
• Vertical stretch or compression depends on $|a|$:
  • If $|a| > 1$, the parabola is narrower than the parent (vertical stretch).
  • If $0 < |a| < 1$, the parabola is wider than the parent (vertical compression).

For example, $f(x) = -2(x - 1)^2 + 5$ takes the parent parabola, shifts it right 1, up 5, stretches it vertically by a factor of 2, and reflects it so it opens downward. The vertex is $(1, 5)$.

Domain and range of quadratic functions

Domain of quadratic functions

The domain is the set of all possible input values. For any quadratic function $f(x) = ax^2 + bx + c$, the domain is all real numbers, since you can square any real number without restriction.

In applied problems, the context might limit the domain. For instance, if $x$ represents time, you'd restrict to $x \geq 0$.

Range of quadratic functions

The range is the set of all possible output values, and it depends on whether the parabola opens up or down.

• If $a > 0$ (opens upward), the vertex is the lowest point. The range is $[k, +\infty)$, where $k$ is the y-coordinate of the vertex.
• If $a < 0$ (opens downward), the vertex is the highest point. The range is $(-\infty, k]$.

In real-world contexts, interpret these based on what the variables represent. If you're modeling the height of a ball over time, the range tells you every possible height the ball reaches, from launch to peak and back down.