We want to draw the graph of the given quadratic function. To do so, we will rewrite it in vertex form,

f (x) = a (x - h)^{2} + k,

where

a,

h,

and

k

are either positive or negative numbers.

g (x) = 0.75 x^{2} - 5 ⇕ g (x) = 0.75 (x - 0)^{2} - 5

To draw the graph, we will follow four steps.

Identify the constants $a,$ $h,$ and $k .$
Plot the vertex $(h, k)$ and draw the axis of symmetry $x = h .$
Plot any point on the curve and its reflection across the axis of symmetry.
Sketch the curve.
Let's get started.

Step $1$
We will first identify the constants $a,$ $h,$ and $k .$ Recall that if $a < 0,$ the parabola will open downwards. Conversely, if $a > 0,$ the parabola will open upwards.
$Vertex Form : Function : f (x) = - a (x - h)^{2} + k g (x) = 0.75 (x - 0)^{2} + (- 5)$
We can see that $a = 0.75,$ $h = 0,$ and $k = - 5 .$ Since $a$ is greater than $0,$ the parabola will open upwards.
Step $2$

Let's now plot the vertex $(h, k)$ and draw the axis of symmetry $x = h .$ Since we already know the values of $h$ and $k,$ we know that the vertex is $(0, - 5) .$ Therefore, the axis of symmetry is the vertical line $x = 0 .$

Step $3$
We will now plot a point on the curve by choosing an $x -$ value and calculating its corresponding $y -$ value. Let's try $x = 4 .$
$g (x) = 0.75 (x)^{2} - 5$
Substitute
$x = 4$
$g (4) = 0.75 (4)^{2} - 5$

▼

Simplify right-hand side

CalcPow
Calculate power
$g (4) = 0.75 (16) - 5$
Multiply
Multiply
$g (4) = 12 - 5$
SubTerm
Subtract term

$g (4) = 7$
When $x = 4,$ we have $y = 7 .$ Thus, the point $(4, 7)$ lies on the curve. Let's plot this point and reflect it across the axis of symmetry.

Note that both points have the same $y -$ coordinate.

Step $4$

Finally, we will sketch the parabola which passes through the three points. Remember not to use a straightedge for this!

Step 1

Step 2

Step 3

Step 4

Step $1$

Step $2$

Step $3$

Step $4$

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