5.1 Quadratic Functions

Quadratic functions describe U-shaped curves called parabolas, and they show up constantly in algebra and beyond. Whether you're modeling the path of a thrown ball or figuring out how to maximize profit, quadratics give you the tools to find key values like maximums, minimums, and zeros.

This section covers the anatomy of parabolas, how to graph them, how to find their extreme values, and how to solve quadratic equations using several techniques.

Quadratic Functions and Parabolas

Features of parabolas

A parabola is a U-shaped curve that's symmetrical about a vertical line. Every parabola is defined by a quadratic function in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

Here are the key features you need to know:

• Vertex: The turning point of the parabola. It's either the lowest point (minimum) or the highest point (maximum), depending on which way the parabola opens. You can find its coordinates using $\left(\frac{-b}{2a},\ f\left(\frac{-b}{2a}\right)\right)$.
• Axis of symmetry: A vertical line that passes through the vertex and splits the parabola into two mirror-image halves. Its equation is $x = \frac{-b}{2a}$.
• Direction of opening: Determined entirely by the leading coefficient $a$.
  • If $a > 0$, the parabola opens upward (like a cup), and the vertex is a minimum.
  • If $a < 0$, the parabola opens downward (like a frown), and the vertex is a maximum.
• Roots (x-intercepts/zeros): The points where the parabola crosses the x-axis. A parabola can have two, one, or zero real roots.
• y-intercept: The point where the parabola crosses the y-axis, which is always $(0, c)$.

Graphing quadratic functions

Starting from the standard form $f(x) = ax^2 + bx + c$, each part of the equation tells you something:

• $a$ controls the direction of opening and how wide or narrow the parabola is. A larger $|a|$ makes the parabola narrower; a smaller $|a|$ makes it wider.
• $b$ influences the horizontal position of the vertex and the axis of symmetry.
• $c$ is the y-intercept, the point where the graph crosses the y-axis.

Steps to graph a quadratic function:

1. Check the sign of $a$ to determine whether the parabola opens up or down.
2. Find the vertex using $\left(\frac{-b}{2a},\ f\left(\frac{-b}{2a}\right)\right)$. Plot this point first.
3. Plot the y-intercept $(0, c)$.
4. Use the axis of symmetry to reflect the y-intercept to the other side of the parabola. For example, if the vertex is at $x = 3$ and the y-intercept is at $x = 0$ (3 units left of the vertex), plot a matching point at $x = 6$ (3 units right).
5. Calculate one or two additional points by plugging x-values into the function, then reflect those as well.
6. Connect the points in a smooth U-shaped curve.

Extrema of quadratic functions

Because a parabola either opens up or opens down, every quadratic function has exactly one extreme value (called an extremum):

• If $a > 0$, the vertex gives you the minimum value of the function. The parabola goes up forever in both directions from there.
• If $a < 0$, the vertex gives you the maximum value. The parabola goes down forever in both directions from there.

To find the extreme value:

1. Calculate the x-coordinate of the vertex: $x = \frac{-b}{2a}$
2. Plug that x-value back into the function to get the y-coordinate. That y-coordinate is the minimum or maximum value.

The extreme value is a global minimum or maximum, meaning no other point on the entire graph is lower (or higher). This is what makes quadratics so useful for optimization: you're guaranteed one best answer.

Applications of quadratic optimization

Whenever a real-world quantity can be modeled by a quadratic function, you can use the vertex to find the optimal (best) outcome.

Steps to solve optimization problems:

1. Read the problem and identify what quantity you need to maximize or minimize.
2. Define your variable(s) and write the quantity to optimize as a quadratic function. You may need to use a constraint (like a fixed perimeter) to reduce to one variable.
3. Find the vertex of that quadratic using $x = \frac{-b}{2a}$, then compute the function value there.
4. Interpret your answer in context, with correct units.

Other common optimization scenarios include maximizing revenue or profit given a price-demand relationship, and finding the maximum height of a projectile.

Solving Quadratic Equations

There are four main tools for solving equations of the form $ax^2 + bx + c = 0$:

Factoring works when the quadratic breaks neatly into two linear factors. For example, $x^2 - 5x + 6 = 0$ factors as $(x - 2)(x - 3) = 0$, giving roots $x = 2$ and $x = 3$. This is the fastest method when it works, but not every quadratic factors cleanly over the integers.

Completing the square rewrites the equation in vertex form $a(x - h)^2 + k = 0$, which you can then solve by isolating the squared term. This technique is also how you derive the quadratic formula itself.

Quadratic formula solves any quadratic equation:

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

This is your go-to when factoring isn't obvious. Just plug in $a$, $b$, and $c$ and simplify.

Discriminant is the expression under the square root: $\Delta = b^2 - 4ac$. It tells you what kind of roots to expect before you finish solving:

• $\Delta > 0$: Two distinct real roots (the parabola crosses the x-axis twice)
• $\Delta = 0$: One repeated real root (the vertex sits exactly on the x-axis)
• $\Delta < 0$: No real roots; two complex roots (the parabola never touches the x-axis)