How to graph a parabola $y = \frac{1}{2} {(x - 3)}^{2} + 5$ $y=1/2(x-3)^2+5$ ?

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How to graph a parabola $y = \frac{1}{2} {(x - 3)}^{2} + 5$ $y=1/2(x-3)^2+5$ ?

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Answer 1 · 2018-04-28T09:31:03

Please read the explanation.

Explanation:

$" "$
$Step 1$ $color(green)("Step 1"$

Construct a data table with input $x$ $color(red)(x$ and **corresponding values for ** $y$ $color(red)y$ :

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

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

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

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

$Step 2$ $color(green)("Step 2"$

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

Useful to analyze the End-behavior of quadratic functions.

enter image source here

$Step 3$ $color(green)("Step 3"$

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

enter image source here

$Step 4$ $color(green)("Step 4"$

$Graph: y = (\frac{1}{2}) \cdot {(x - 3)}^{2}$ $color(red)("Graph: " y=(1/2)*(x-3)^2$

enter image source here

$Step 5$ $color(green)("Step 5"$

$Graph: y = (\frac{1}{2}) \cdot {(x - 3)}^{2} + 5$ $color(red)("Graph: " y=(1/2)*(x-3)^2+5$

enter image source here

$Step 6$ $color(green)("Step 6"$

View all the graphs together:

enter image source here

What we observe ?

$y = f (x) = (\frac{1}{2}) \cdot {(x - 3)}^{2} + 5$ $color(brown)(y=f(x)=(1/2)*(x-3)^2+5$

General form: $y = f (x) = a {(x - h)}^{2} + k$ $color(blue)(y=f(x)=a(x-h)^2+k$ , Vertex: $(h . k)$ $color(green)((h.k)$
On the graph in $Step 4$ $color(green)("Step 4")$ we have $V e r t e x : (3, 5)$ $color(blue)(Vertex: (3,5)$
Graph Opens up, as the $x^{2}$ $x^2$ term is positive.
Parabolic curve is expanded outward, as $0 < a < 1$ $color(red)(0 < a < 1$
$x = h$ $x=h$ , and in our problem $x = 3$ $x=3$ is the Axis of Symmetry
$h = 3$ $h=3$ indicates the Horizontal Shift
$k = 5$ $k=5$ indicates the Vertical Shift

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How to graph a parabola $y = \frac{1}{2} {(x - 3)}^{2} + 5$ $y=1/2(x-3)^2+5$ ?

1 Answer

Explanation:

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