Skill: Recognize quadratic relations from graphs

A quadratic relation has a graph shaped like a parabola. To recognize one, check for:

• One line of symmetry, called the axis of symmetry
• One turning point, called the vertex
• A smooth, consistent curve
• An opening upward or downward

A vertical parabola can be written as

$$y=ax^2+bx+c$$

If $a>0$, it opens upward. If $a<0$, it opens downward.

Example: Suppose a graph contains the points

$$(-2,3),\quad (-1,0),\quad (0,-1),\quad (1,0),\quad (2,3).$$

Step 1: Look at the overall shape.
The graph decreases until it reaches $(0,-1)$ and then increases. It has a smooth, U-shaped curve, so it could be a parabola.

Step 2: Find the vertex.
The lowest point is

$$(0,-1).$$

This is the vertex.

Step 3: Check for symmetry.
The points on either side of $x=0$ match:

• At $x=-1$ and $x=1$, $y=0$
• At $x=-2$ and $x=2$, $y=3$

Therefore, the axis of symmetry is

$$x=0.$$

Step 4: Identify the intercepts.

• The graph crosses the $y$-axis at $(0,-1)$, so the $y$-intercept is $-1$.
• The graph crosses the $x$-axis at $(-1,0)$ and $(1,0)$, so the zeros are $x=-1$ and $x=1$.

Step 5: State the conclusion.
The graph is a quadratic relation because it is a smooth, U-shaped curve with one vertex and one axis of symmetry. It opens upward, so its leading coefficient is positive.

It matches the equation

$$y=x^2-1.$$

A straight line is linear, and a V-shaped graph is an absolute-value relation. Neither has the smooth, curved shape and single vertex of this parabola.