Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a parabola “opens upward” when the lowest point on the graph is the vertex (where the graph changes direction), and “opens downward” when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written in standard form $ax^2 + bx+ c$ tells us something important about the graph that represents it.

The graph of $y=x^2$ is a parabola opening upward with vertex at $(0,0)$. Adding a constant term 5 gives $y = x^2 + 5$ and raises the graph by 5 units. Subtracting 4 from $x^2$ gives $y=x^2-4$ and moves the graph 4 units down.

$x$	-3	-2	-1	0	1	2	3
$x^2$	9	4	1	0	1	4	9
$x^2+5$	14	9	6	5	6	9	14
$x^2-4$	5	0	-3	-4	-3	0	5

A table of values can help us see that adding 5 to $x^2$ increases all the output values of $y = x^2$ by 5, which explains why the graph moves up 5 units. Subtracting 4 from $x^2$ decreases all the output values of $y = x^2$ by 4, which explains why the graph shifts down by 4 units.

In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. An expression with no constant term (such as $x^2$ or $x^2 +9x$) means that the constant term is 0, so the $y$-intercept of the graph is on the $x$-axis. It’s not shifted up or down relative to the $x$-axis.

The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient of the squared term in $y = x^2$ is 1. Its graph is a parabola that opens upward.

• Multiplying $x^2$ by a number greater than 1 makes the graph steeper, so the parabola is narrower than that representing $x^2$.
• Multiplying $x^2$ by a number less than 1 but greater than 0 makes the graph less steep, so the parabola is wider than that representing $x^2$.
• Multiplying $x^2$ by a number less than 0 makes the parabola open downward.

Coordinate plane, 4 graphs of quadratic functions, all with the maximum or minimum at the origin. First, y = 2 x squared, opens up. Second, y = x squared, opens up but wider than the first. Third, y = fraction 1 over 2 x squared, opens up wider than the first 2. Fourth, y = negative 2 x squared, opens down.

$x$	-3	-2	-1	0	1	2	3
$x^2$	9	4	1	0	1	4	9
$2x^2$	18	8	2	0	2	8	18
$\text-2x^2$	-18	-8	-2	0	-2	-8	-18

If we compare the output values of $2x^2$ and $\text-2x^2$, we see that they are opposites, which suggests that one graph would be a reflection of the other across the $x$-axis.