QuestionFind the axis of symmetry of $f(x)=x^2+5$ and the graph of $y=-36-3x+3x^2$ .

Studdy Solution

STEP 1

Assumptions
1. The quadratic equation for the axis of symmetry is given as $f(x) = x^2 + 5$ .
2. The quadratic equation for the graph characteristics is given as $y = -36 - 3x + 3x^2$ .
3. The axis of symmetry for a quadratic equation in standard form $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$ .
4. The direction in which a quadratic equation opens is determined by the sign of the coefficient $a$ in the standard form $ax^2 + bx + c$ . If $a > 0$ , it opens upwards; if $a < 0$ , it opens downwards.
5. A quadratic equation has a maximum if it opens downwards and a minimum if it opens upwards.

STEP 2

First, we will find the axis of symmetry for the quadratic equation $f(x) = x^2 + 5$ .
The standard form of a quadratic equation is $ax^2 + bx + c$ . Here, $a = 1$ and $b = 0$ .

STEP 3

Use the formula for the axis of symmetry $x = -\frac{b}{2a}$ to find the axis of symmetry for $f(x)$ .
$x = -\frac{0}{2 \cdot 1}$

STEP 4

Calculate the axis of symmetry.
$x = -\frac{0}{2} = 0$
The axis of symmetry for the quadratic equation $f(x) = x^2 + 5$ is $x = 0$ .

STEP 5

Now, we will determine the characteristics of the graph of the quadratic equation $y = -36 - 3x + 3x^2$ .
Rewrite the equation in standard form by rearranging the terms in descending order of the power of $x$ .
$y = 3x^2 - 3x - 36$

STEP 6

Identify the coefficient $a$ in the standard form $ax^2 + bx + c$ . Here, $a = 3$ .

STEP 7

Since $a = 3$ is positive, the graph of the quadratic equation opens upwards.

STEP 8

Because the graph opens upwards, it has a minimum point.

STEP 9

Combine the information from steps 7 and 8 to determine the characteristic of the graph.
The graph of the quadratic equation $y = -36 - 3x + 3x^2$ has a minimum and opens up.

STEP 10

Now, we can answer the questions based on the calculations.
For the axis of symmetry of the quadratic equation $f(x) = x^2 + 5$ , the correct answer is: a. $x = 0$
For the graph of the quadratic equation $y = -36 - 3x + 3x^2$ , the correct answer is: d. Has minimum and opens up

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms