# How to graph a parabola y=1/2(x-3)^2+5?

Apr 28, 2018

#### Explanation:

$\text{ }$
color(green)("Step 1"

Construct a data table with input color(red)(x and corresponding values for $\textcolor{red}{y}$:

This table will help immensely in understanding the End Behavior of the given function: color(blue)(y=f(x)=(1/2)*(x-3)^2+5

color(red)(x: -5<=x<=5 [ Col 1 ]

Draw graphs for $y = {x}^{2}$, $y = {\left(x - 3\right)}^{2}$, $y = \left(\frac{1}{2}\right) {\left(x - 3\right)}^{2}$, and finally $y = \left(\frac{1}{2}\right) {\left(x - 3\right)}^{2} + 5$

Find Vertices, x-intercept and y-intercept, if any, for all the graphs.

color(green)("Step 2"

color(red)("Graph: " y=x^2 .....Parent Quadratic Function

Useful to analyze the End-behavior of quadratic functions.

color(green)("Step 3"

color(red)("Graph: " y=(x-3)^2

color(green)("Step 4"

color(red)("Graph: " y=(1/2)*(x-3)^2

color(green)("Step 5"

color(red)("Graph: " y=(1/2)*(x-3)^2+5

color(green)("Step 6"

View all the graphs together:

What we observe ?

color(brown)(y=f(x)=(1/2)*(x-3)^2+5

1. General form: color(blue)(y=f(x)=a(x-h)^2+k, Vertex: color(green)((h.k)

2. On the graph in $\textcolor{g r e e n}{\text{Step 4}}$ we have color(blue)(Vertex: (3,5)

3. Graph Opens up, as the ${x}^{2}$ term is positive.

4. Parabolic curve is expanded outward, as color(red)(0 < a < 1

5. $x = h$, and in our problem $x = 3$ is the Axis of Symmetry

6. $h = 3$ indicates the Horizontal Shift

7. $k = 5$ indicates the Vertical Shift