Reading Quadratic Functions

The idea in one sentence

A quadratic function graphs as a parabola, and its form can reveal the turning point, direction, and zeros.

Start with the parent function

The simplest quadratic is:

\[y = x^2\]

Its graph is a U-shaped curve with vertex $(0,0)$. Because the coefficient of $x^2$ is positive, it opens upward.

Read vertex form

Vertex form is:

\[y = a(x-h)^2 + k\]

The vertex is $(h,k)$. For $y = 2(x-3)^2 - 4$:

• the vertex is $(3,-4)$;
• the parabola opens upward because $a=2$ is positive;

• it is narrower than $y=x^2$ because $

  a

  >1$.

Watch the sign inside the parentheses: $(x-3)$ gives $h=3$, while $(x+3)$ gives $h=-3$.

Find the zeros

Zeros are the $x$-values where $y=0$. If a quadratic is factored:

\[y=(x-2)(x+5)\]

Set each factor equal to zero. The zeros are $x=2$ and $x=-5$. Those are the points where the graph crosses the $x$-axis.

Common mix-ups

• Reading the vertex sign incorrectly in vertex form.
• Assuming every parabola crosses the $x$-axis twice. It may cross twice, touch once, or never cross.
• Looking only at $a$ for the vertex. The coefficient controls direction and width, not location.

Check yourself

For $y=-3(x+1)^2+6$, name the vertex and say whether the parabola opens up or down.

The vertex is $(-1,6)$, and the parabola opens down because $a=-3$.