Math / Algebra

Questionsider the following quadratic function. $g(x)=-3 x^{2}+24 x-41$ (a) Write the equation in the form $g(x)=a(x-h)^{2}+k$ . Then give the vertex of its graph.
Writing in the form specified: $g(x)=$ $\square$ II
Vertex: $\square$ $\square$ D)

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a quadratic function from its standard form to its vertex form and then find the coordinates of its vertex! Watch out! Don't forget to keep track of those negative signs when completing the square.
It's a super common trip-up!

STEP 2

1. Prepare for Vertex Form
2. Complete the Square
3. Identify the Vertex

STEP 3

Let's factor out the coefficient of $x^2$ , which is $-3$ , from the first two terms of our function $g(x) = -3x^2 + 24x - 41$ .
This sets us up perfectly to complete the square.
This gives us: $g(x) = -3(x^2 - 8x) - 41$ Why did we do this?
Because completing the square is easiest when the coefficient of $x^2$ is $1$ .

STEP 4

Now, let's find our magic number!
Take half of the coefficient of our $x$ term inside the parentheses (which is $-8$ ), and square it! $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$ So, our magic number is 16.

STEP 5

We'll add and subtract this magic number inside the parentheses.
Remember, adding and subtracting the same number is like adding zero, so it doesn't change the value of our function! $g(x) = -3(x^2 - 8x + 16 - 16) - 41$

STEP 6

Notice that $x^2 - 8x + 16$ is a perfect square trinomial.
It factors to $(x-4)^2$ .
So, we can rewrite our function as: $g(x) = -3((x-4)^2 - 16) - 41$

STEP 7

Now, let's distribute the $-3$ and simplify: $g(x) = -3(x-4)^2 -3 \cdot (-16) - 41$ $g(x) = -3(x-4)^2 + 48 - 41$ $g(x) = -3(x-4)^2 + 7$

STEP 8

Our function is now in vertex form, $g(x) = a(x-h)^2 + k$ , where the vertex is given by the point $(h, k)$ .
In our case, $h = 4$ and $k = 7$ , so our vertex is at $(4, 7)$ .

STEP 9

Writing in the form specified: $g(x) = -3(x-4)^2 + 7$ Vertex: $(4, 7)$

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Questionsider the following quadratic function. $g(x)=-3 x^{2}+24 x-41$ (a) Write the equation in the form $g(x)=a(x-h)^{2}+k$ . Then give the vertex of its graph.
Writing in the form specified: $g(x)=$ $\square$ II
Vertex: $\square$ $\square$ D)

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a quadratic function from its standard form to its vertex form and then find the coordinates of its vertex! Watch out! Don't forget to keep track of those negative signs when completing the square.
It's a super common trip-up!

STEP 2

1. Prepare for Vertex Form
2. Complete the Square
3. Identify the Vertex

STEP 3

STEP 4

Now, let's find our magic number!
Take half of the coefficient of our $x$ term inside the parentheses (which is $-8$ ), and square it! $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$ So, our magic number is 16.

STEP 5

We'll add and subtract this magic number inside the parentheses.
Remember, adding and subtracting the same number is like adding zero, so it doesn't change the value of our function! $g(x) = -3(x^2 - 8x + 16 - 16) - 41$

STEP 6

Notice that $x^2 - 8x + 16$ is a perfect square trinomial.
It factors to $(x-4)^2$ .
So, we can rewrite our function as: $g(x) = -3((x-4)^2 - 16) - 41$

STEP 7

Now, let's distribute the $-3$ and simplify: $g(x) = -3(x-4)^2 -3 \cdot (-16) - 41$ $g(x) = -3(x-4)^2 + 48 - 41$ $g(x) = -3(x-4)^2 + 7$

STEP 8

Our function is now in vertex form, $g(x) = a(x-h)^2 + k$ , where the vertex is given by the point $(h, k)$ .
In our case, $h = 4$ and $k = 7$ , so our vertex is at $(4, 7)$ .

STEP 9

Writing in the form specified: $g(x) = -3(x-4)^2 + 7$ Vertex: $(4, 7)$

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Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a quadratic function from its standard form to its vertex form and then find the coordinates of its vertex! Watch out! Don't forget to keep track of those negative signs when completing the square.It's a super common trip-up!

STEP 2

1. Prepare for Vertex Form2. Complete the Square3. Identify the Vertex

STEP 3

STEP 4

Now, let's find our **magic number**!Take half of the coefficient of our xxx term inside the parentheses (which is −8-8−8), and square it! (−82)2=(−4)2=16 \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 (2−8​)2=(−4)2=16 So, our magic number is **16**.

STEP 5

We'll add and subtract this magic number inside the parentheses.Remember, adding and subtracting the same number is like adding zero, so it doesn't change the value of our function! g(x)=−3(x2−8x+16−16)−41 g(x) = -3(x^2 - 8x + 16 - 16) - 41 g(x)=−3(x2−8x+16−16)−41

STEP 6

Notice that x2−8x+16x^2 - 8x + 16x2−8x+16 is a perfect square trinomial.It factors to (x−4)2(x-4)^2(x−4)2.So, we can rewrite our function as: g(x)=−3((x−4)2−16)−41 g(x) = -3((x-4)^2 - 16) - 41 g(x)=−3((x−4)2−16)−41

STEP 7

STEP 8

Our function is now in vertex form, g(x)=a(x−h)2+kg(x) = a(x-h)^2 + kg(x)=a(x−h)2+k, where the vertex is given by the point (h,k)(h, k)(h,k).In our case, h=4h = 4h=4 and k=7k = 7k=7, so our vertex is at (4,7)(4, 7)(4,7).

STEP 9

Writing in the form specified: g(x)=−3(x−4)2+7g(x) = -3(x-4)^2 + 7g(x)=−3(x−4)2+7 Vertex: (4,7)(4, 7)(4,7)

Get Studdy

Studdy solves anything!

Start learning now

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

What is this asking? We're asked to rewrite a quadratic function from its standard form to its vertex form and then find the coordinates of its vertex! Watch out! Don't forget to keep track of those negative signs when completing the square.
It's a super common trip-up!

1. Prepare for Vertex Form
2. Complete the Square
3. Identify the Vertex

Now, let's find our magic number!
Take half of the coefficient of our $x$ term inside the parentheses (which is $-8$ ), and square it! $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$ So, our magic number is 16.

We'll add and subtract this magic number inside the parentheses.
Remember, adding and subtracting the same number is like adding zero, so it doesn't change the value of our function! $g(x) = -3(x^2 - 8x + 16 - 16) - 41$

Notice that $x^2 - 8x + 16$ is a perfect square trinomial.
It factors to $(x-4)^2$ .
So, we can rewrite our function as: $g(x) = -3((x-4)^2 - 16) - 41$

Our function is now in vertex form, $g(x) = a(x-h)^2 + k$ , where the vertex is given by the point $(h, k)$ .
In our case, $h = 4$ and $k = 7$ , so our vertex is at $(4, 7)$ .

Writing in the form specified: $g(x) = -3(x-4)^2 + 7$ Vertex: $(4, 7)$