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:

$f(x) = 2x^2 + 12x + 17$

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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.