Use the vertex (h,k) and a point on the graph (x,y)

 to find the standard form of the quadratic function.

 

(h,k)=(−3,−1)
, (x,y)=(−4,1)

To find the standard form of the quadratic function using the vertex \((h, k) = (-3, -1)\) and a point on the graph \((x, y) = (-4, 1)\), we follow these steps:

### Step 1: Use the vertex form of a quadratic function
The vertex form of a quadratic function is:

\[
f(x) = a(x - h)^2 + k
\]

Where:
- \(a\) is a constant to be determined.
- \((h, k)\) is the vertex of the parabola.

Substitute the given vertex \((h, k) = (-3, -1)\) into the vertex form:

\[
f(x) = a(x + 3)^2 - 1
\]

### Step 2: Substitute the point \((x, y) = (-4, 1)\) into the equation
Now, use the point \((x, y) = (-4, 1)\) to find the value of \(a\). Substitute \(x = -4\) and \(y = 1\) into the equation:

\[
1 = a(-4 + 3)^2 - 1
\]

Simplify the equation:

\[
1 = a(1)^2 - 1
\]

\[
1 = a - 1
\]

### Step 3: Solve for \(a\)
Add 1 to both sides:

\[
2 = a
\]

Thus, \(a = 2\).

### Step 4: Write the standard form
Now that we know \(a = 2\), substitute this value into the equation:

\[
f(x) = 2(x + 3)^2 - 1
\]

This is the vertex form of the quadratic function. To express this in standard form, we expand the equation:

\[
f(x) = 2(x^2 + 6x + 9) - 1
\]

\[
f(x) = 2x^2 + 12x + 18 - 1
\]

\[
f(x) = 2x^2 + 12x + 17
\]

### Final Answer:
The standard form of the quadratic function is:

\[
f(x) = 2x^2 + 12x + 17
\]

---

Would you like a more detailed explanation on any of the steps?  

Here are 5 related questions for further practice:

1. How do you convert a quadratic function from standard form to vertex form?
2. How can you find the axis of symmetry of a quadratic function in standard form?
3. What is the importance of the vertex in the graph of a quadratic function?
4. How do you determine if a quadratic function opens upward or downward?
5. How do you calculate the y-intercept of a quadratic function?

**Tip:** Always double-check your calculations when solving for \(a\) using a point, as any small mistake can affect the entire equation.

Use the vertex (h,k) and a point on the graph (x,y) to find the standard form of the quadratic function. (h,k)=(-3,-1), (x,y)=(-4,1)

Learn how to find the standard form of a quadratic function using the vertex (-3, -1) and a point on the graph (-4, 1). This step-by-step solution covers important algebraic concepts such as vertex form and standard form. Ideal for high school students in Grades 9-11.

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