Lesson 27: Connecting the Parabola with the Quadratic Function

New Concept

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

What’s next

Next, you'll bring these functions to life by graphing their parabolic curves and identifying the key features that unlock real-world applications.

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$ and $a$, $b$, and $c$ are real numbers. This is also known as the standard form.

To write $y + 3x = 8 - 2x^2$ in standard form, rearrange the terms: $y = -2x^2 - 3x + 8$.
To convert $f(x) = 3(x - 2)^2 + 5$ to standard form, expand $(x-2)^2$, then distribute and combine terms: $f(x) = 3(x^2 - 4x + 4) + 5 = 3x^2 - 12x + 17$.

This is the official 'dress code' for quadratic functions. Every term lines up neatly by its power of x, from $x^2$ down to the constant. Arranging it this way makes it much easier to analyze the function's graph and behavior, like finding the y-intercept, which is just 'c'! It turns messy equations into something we can work with.

The graph of every quadratic function is a U-shaped curve called a parabola. The vertex of the parabola is its lowest (or highest) point and indicates where the curve changes direction.

The graph of the parent function $f(x) = x^2$ is a parabola that opens upward with its vertex at the origin $(0, 0)$.
The graph of $f(x) = -x^2 + 3$ is a parabola that opens downward, with its vertex at $(0, 3)$.
For $f(x) = (x-4)^2$, the graph is a parabola shifted right, with its vertex at $(4, 0)$.

Every quadratic function graph is a 'U'-shaped curve called a parabola! It can open upwards like a smile or downwards like a frown. The most important spot is the vertex, the absolute bottom or top of the curve. It's the turning point where the whole graph changes direction, giving it that perfect, cool symmetry.

The axis of symmetry is a vertical line that divides a parabola into two congruent mirror images. The equation for this line is $x = -\frac{b}{2a}$.

For the function $f(x) = x^2 - x - 6$, where $a=1$ and $b=-1$, the axis of symmetry is $x = -\frac{(-1)}{2(1)} = \frac{1}{2}$.
Given $f(x) = 2x^2 - 12x + 5$, where $a=2$ and $b=-12$, the axis of symmetry is $x = -\frac{(-12)}{2(2)} = \frac{12}{4} = 3$.

Imagine folding a paper parabola perfectly in half; the crease line is its axis of symmetry! It's a vertical line that cuts right through the vertex, making the left side a perfect mirror image of the right. The magic formula $x = -\frac{b}{2a}$ instantly gives you this line for any quadratic in standard form.

The zeros of a quadratic function are the values of $x$ for which the function equals $0$. On a graph, the zeros are the $x$-intercepts, or the points where the graph intersects the $x$-axis.

For the function $f(x) = x^2 - 9$, the zeros are found by setting $x^2 - 9 = 0$, which gives $x = 3$ and $x = -3$.
The function $f(x) = x^2 - 5x + 4$ has zeros at $x=1$ and $x=4$, because $(1)^2 - 5(1) + 4 = 0$ and $(4)^2 - 5(4) + 4 = 0$.

The 'zeros' are where the parabola party hits the ground! They are the special x-values that make the function equal to zero, which means they are the exact points where the graph crosses the x-axis. Finding the zeros helps you solve real-world problems, like figuring out how long it takes for a launched rocket to land.