Quadratic Parent Functions Solutions

Polynomial Parent Functions
Type	Linear	Quadratic	Cubic
Parent equation	y = x	$y=x^2$
Degree	1	2
Domain and Range	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}\right\} \end{gathered}$$	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\} \end{gathered}$$
Quadrants	Q1 and Q3	Q1 and Q2
Because	the x- and y-coordinates will always have the same value	the y-coordinates will always be positive, because any number squared is positive
Other Characteristics	slope $m=1$	parabola, u-shaped symmetric graph Axis of symmetry is $x=0,$ the y-axis
Intersects graph	Polynomial parent functions intersect the coordinate plane at the origin: (0, 0)

Note

The parabola changes directions at the origin because of the direction of the end behavior.

Example 4

Graph the quadratic parent function. Name the domain and range in interval and set-builder notation. Then name the end behavior of the graph.

End behavior

As $x\to +\infty , f\left(x\right)\to \underline{ +\infty }$, and as $x\to –\infty , \underline{ f\left(x\right)\to +\infty }$

Note

Instruction on graphing parabolas by hand without using a table can be found in Algebra 1: Principles of Secondary Mathematics.