Lesson 2: Graphing f (x) = ax² + c

Property

The graph of a quadratic function $f(x) = ax^2 + c$ combines the effects of both the coefficient $a$ and the constant term $c$.
For $f(x) = ax^2 + c$:

• The parabola opens upward if $a > 0$ and downward if $a < 0$
• The magnitude of $|a|$ determines the width: larger values make it narrower, smaller values make it wider
• The constant $c$ shifts the graph vertically: upward if $c > 0$, downward if $c < 0$
• The vertex is located at $(0, c)$

Examples

Property

Compared to the graph of $y = x^2$, the graph of $y = x^2 + c$

• is shifted upward by $c$ units if $c > 0$.
• is shifted downward by $|c|$ units if $c < 0$.

Examples

• The graph of $y = x^2 + 5$ is the basic parabola shifted 5 units up. Its vertex is located at $(0, 5)$.
• The graph of $y = x^2 - 3$ is the basic parabola shifted 3 units down. Its vertex is located at $(0, -3)$.
• The graph of $y = -x^2 + 1$ is an upside-down parabola that has been shifted 1 unit up. Its vertex is at $(0, 1).

Property

The $x$-intercepts of the graph of

are the solutions of the equation

To find the $x$-intercepts of a parabola in the form $f(x) = ax^2 + c$, set $f(x) = 0$ and solve for $x$.

Examples