Introduction to Quadratic Functions
Define quadratic functions as functions of the form f(x) = ax² + bx + c. Identify their parabolic graphs and recognize the effect of changing a on the direction and width of the parabola.
A lesson of Algebra 1, in Quadratic Functions, about 30 minutes.
By the end of this lesson, you will be able to recognize a quadratic function, name $a$, $b$ and $c$, predict which way its parabola opens and how wide it is, and read the $y$-intercept without any work.
Degree Two Changes the Shape Entirely
A linear function has a constant rate of change. A quadratic does not, and that single difference produces a curve.
$$f(x) = ax^2 + bx + c \qquad a \neq 0$$
The $a \neq 0$ is not a technicality. Set $a = 0$ and the $x^2$ term disappears, leaving $bx + c$, which is a line. The whole of this unit exists because $a$ is not zero.
Look at what happens to the outputs as $x$ climbs by one each time, for $f(x) = x^2$:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|
| $f(x)$ | $4$ | $1$ | $0$ | $1$ | $4$ | $9$ |
| change | $-3$ | $-1$ | $+1$ | $+3$ | $+5$ |
The changes are not constant, so the graph cannot be straight. But look again: the changes themselves go up by $2$ every time. The second differences are constant, and that is the fingerprint of a quadratic, exactly as constant first differences are the fingerprint of a line.
The Parabola, and Its Symmetry
The graph of every quadratic is a parabola: a single smooth U, opening either up or down, with one turning point.
$$f(x) = x^2 \qquad f(-3) = 9 \qquad f(3) = 9$$
Inputs either side of the turning point give the same output, because squaring loses the sign. So a parabola is symmetric about a vertical line through its turning point, and that symmetry is the most useful thing about it: find one point and you have a second one free.
Three words to have:
- The vertex is the turning point, the lowest point of an upward parabola or the highest of a downward one
- The axis of symmetry is the vertical line through the vertex
- The zeros or roots are where the parabola crosses the $x$-axis, which are the solutions of $ax^2 + bx + c = 0$
A parabola can cross the $x$-axis twice, touch it once, or miss it entirely, which is why a quadratic equation can have two solutions, one, or none.
Sort It: Quadratic or Not
The fifth one is the trap. $a = 0$ is disqualifying, no matter how the expression is written.
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