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Question 2 (4 points)

Write the equation of the quadratic function in vertex and standard form whose vertex is at [tex]\((-1, 6)\)[/tex] and that passes through [tex]\((2, 24)\)[/tex].

Sagot :

GinnyAnswer
To find the equation of the quadratic function in both vertex and standard form given that the vertex is at [tex]\((-1, 6)\)[/tex] and the parabola passes through the point [tex]\((2, 24)\)[/tex], we can follow these steps:

### Step-by-Step Solution:

#### 1. Understanding the Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is:

[tex]\[ y = a(x - h)^2 + k \][/tex]

where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Here, the vertex [tex]\((h, k)\)[/tex] is given as [tex]\((-1, 6)\)[/tex], so [tex]\(h = -1\)[/tex] and [tex]\(k = 6\)[/tex].

#### 2. Plugging the Vertex into the Equation:
Substituting the vertex values into the vertex form equation, we get:

[tex]\[ y = a(x - (-1))^2 + 6 \][/tex]
[tex]\[ y = a(x + 1)^2 + 6 \][/tex]

#### 3. Using the Given Point to Find [tex]\(a\)[/tex]:
We know that the function passes through the point [tex]\((2, 24)\)[/tex]. Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 24\)[/tex] into the equation, we get:

[tex]\[ 24 = a(2 + 1)^2 + 6 \][/tex]
[tex]\[ 24 = a \cdot 3^2 + 6 \][/tex]
[tex]\[ 24 = 9a + 6 \][/tex]

Solving for [tex]\(a\)[/tex]:

[tex]\[ 24 - 6 = 9a \][/tex]
[tex]\[ 18 = 9a \][/tex]
[tex]\[ a = 2 \][/tex]

So, we have found that [tex]\(a = 2\)[/tex].

#### 4. Writing the Vertex Form:
Now that we have [tex]\(a = 2\)[/tex], [tex]\(h = -1\)[/tex], and [tex]\(k = 6\)[/tex], the vertex form of the quadratic function is:

[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]

#### 5. Converting Vertex Form to Standard Form:
The standard form of a quadratic function is:

[tex]\[ y = ax^2 + bx + c \][/tex]

To convert [tex]\( y = 2(x + 1)^2 + 6 \)[/tex] to standard form, we need to expand it:

[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]
[tex]\[ y = 2(x^2 + 2x + 1) + 6 \][/tex]
[tex]\[ y = 2x^2 + 4x + 2 + 6 \][/tex]
[tex]\[ y = 2x^2 + 4x + 8 \][/tex]

So, the quadratic function in standard form is:

[tex]\[ y = 2x^2 + 4x + 8 \][/tex]

#### Summary:
- Vertex Form: [tex]\( y = 2(x + 1)^2 + 6 \)[/tex]
- Standard Form: [tex]\( y = 2x^2 + 4x + 8 \)[/tex]

These are the equations of the quadratic function that satisfy the given conditions.
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