Select The Quadratic Function With A Graph That Has The Following Features:- $x$-intercept At $(8,0)$- $y$-intercept At $(0,-32)$- Maximum Value At $(6,4)$- Axis Of Symmetry At $x=6$A.

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Understanding Quadratic Functions

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Quadratic functions are a type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These functions have a parabolic shape and can have various features such as $x$-intercepts, $y$-intercepts, maximum or minimum values, and an axis of symmetry. In this article, we will discuss how to select the quadratic function with a graph that has specific features.

Features of the Graph

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The graph of a quadratic function can have several features, including:

• $x$-intercept: The point where the graph intersects the $x$-axis. This occurs when $y = 0$.
• $y$-intercept: The point where the graph intersects the $y$-axis. This occurs when $x = 0$.
• Maximum or minimum value: The highest or lowest point on the graph.
• Axis of symmetry: A vertical line that passes through the maximum or minimum value and divides the graph into two symmetrical parts.

Given Features

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The given features of the graph are:

• $x$-intercept at $(8,0)$: This means that the graph intersects the $x$-axis at the point $(8,0)$.
• $y$-intercept at $(0,-32)$: This means that the graph intersects the $y$-axis at the point $(0,-32)$.
• Maximum value at $(6,4)$: This means that the highest point on the graph is at the point $(6,4)$.
• Axis of symmetry at $x=6$: This means that the vertical line $x=6$ passes through the maximum value and divides the graph into two symmetrical parts.

Selecting the Quadratic Function

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To select the quadratic function with a graph that has the given features, we need to use the following steps:

Step 1: Write the General Form of the Quadratic Function

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The general form of a quadratic function is $f(x) = ax^2 + bx + c$. We can start by writing this function in the general form.

Step 2: Use the $x$-intercept to Find the Value of $c$

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The $x$-intercept is the point where the graph intersects the $x$-axis. This occurs when $y = 0$. We can substitute the $x$-intercept into the general form of the function to find the value of $c$.

Step 3: Use the $y$-intercept to Find the Value of $a$

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The $y$-intercept is the point where the graph intersects the $y$-axis. This occurs when $x = 0$. We can substitute the $y$-intercept into the general form of the function to find the value of $a$.

Step 4: Use the Maximum Value to Find the Value of $b$

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The maximum value is the highest point on the graph. We can use this point to find the value of $b$.

Step 5: Use the Axis of Symmetry to Find the Value of $a$ and $b$

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The axis of symmetry is a vertical line that passes through the maximum value and divides the graph into two symmetrical parts. We can use this line to find the values of $a$ and $b$.

Solving for $a$, $b$, and $c$

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Let's solve for $a$, $b$, and $c$ using the given features.

Step 1: Write the General Form of the Quadratic Function

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$f(x) = ax^2 + bx + c$

Step 2: Use the $x$-intercept to Find the Value of $c$

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The $x$-intercept is $(8,0)$. We can substitute this point into the general form of the function to find the value of $c$.

$0 = a(8)^2 + b(8) + c$

$0 = 64a + 8b + c$

Step 3: Use the $y$-intercept to Find the Value of $a$

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The $y$-intercept is $(0,-32)$. We can substitute this point into the general form of the function to find the value of $a$.

$-32 = a(0)^2 + b(0) + c$

$-32 = c$

Step 4: Use the Maximum Value to Find the Value of $b$

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The maximum value is $(6,4)$. We can use this point to find the value of $b$.

$4 = a(6)^2 + b(6) - 32$

$4 = 36a + 6b - 32$

$36a + 6b = 36$

Step 5: Use the Axis of Symmetry to Find the Value of $a$ and $b$

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The axis of symmetry is $x=6$. We can use this line to find the values of $a$ and $b$.

Since the axis of symmetry is $x=6$, we know that the vertex of the parabola is at the point $(6,4)$. We can use this information to find the values of $a$ and $b$.

The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. We can substitute the vertex into this form to find the values of $a$ and $b$.

$f(x) = a(x-6)^2 + 4$

We can expand this expression to find the values of $a$ and $b$.

$f(x) = a(x^2 - 12x + 36) + 4$

$f(x) = ax^2 - 12ax + 36a + 4$

We can compare this expression to the general form of the quadratic function to find the values of $a$ and $b$.

$ax^2 + bx + c = ax^2 - 12ax + 36a + 4$

We can equate the coefficients of the two expressions to find the values of $a$ and $b$.

$a = a$

$-12a = b$

$36a + 4 = c$

We can solve for $a$, $b$, and $c$ using these equations.

Solving for $a$

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We can solve for $a$ using the equation $36a + 4 = c$.

$36a + 4 = -32$

$36a = -36$

$a = -1$

Solving for $b$

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We can solve for $b$ using the equation $-12a = b$.

$-12a = b$

$-12(-1) = b$

$b = 12$

Solving for $c$

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We can solve for $c$ using the equation $36a + 4 = c$.

$36a + 4 = c$

$36(-1) + 4 = c$

$c = -32$

The Quadratic Function

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We can now substitute the values of $a$, $b$, and $c$ into the general form of the quadratic function to find the final answer.

$f(x) = ax^2 + bx + c$

$f(x) = -x^2 + 12x - 32$

The final answer is $\boxed{-x^2 + 12x - 32}$.

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Understanding Quadratic Functions

---

Quadratic functions are a type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These functions have a parabolic shape and can have various features such as $x$-intercepts, $y$-intercepts, maximum or minimum values, and an axis of symmetry.

Frequently Asked Questions

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Q: What is a quadratic function?

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A quadratic function is a type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What are the features of a quadratic function?

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The features of a quadratic function include:

• $x$-intercept: The point where the graph intersects the $x$-axis. This occurs when $y = 0$.
• $y$-intercept: The point where the graph intersects the $y$-axis. This occurs when $x = 0$.
• Maximum or minimum value: The highest or lowest point on the graph.
• Axis of symmetry: A vertical line that passes through the maximum or minimum value and divides the graph into two symmetrical parts.

Q: How do I find the value of $a$, $b$, and $c$ in a quadratic function?

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To find the value of $a$, $b$, and $c$ in a quadratic function, you can use the following steps:

1. Use the $x$-intercept to find the value of $c$: Substitute the $x$-intercept into the general form of the function to find the value of $c$.
2. Use the $y$-intercept to find the value of $a$: Substitute the $y$-intercept into the general form of the function to find the value of $a$.
3. Use the maximum or minimum value to find the value of $b$: Use the maximum or minimum value to find the value of $b$.
4. Use the axis of symmetry to find the value of $a$ and $b$: Use the axis of symmetry to find the values of $a$ and $b$.

Q: How do I write a quadratic function in vertex form?

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To write a quadratic function in vertex form, you can use the following steps:

1. Identify the vertex: Identify the vertex of the parabola, which is the point $(h,k)$.
2. Write the function in vertex form: Write the function in the form $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex.

Q: How do I find the axis of symmetry of a quadratic function?

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To find the axis of symmetry of a quadratic function, you can use the following steps:

1. Identify the vertex: Identify the vertex of the parabola, which is the point $(h,k)$.
2. Write the function in vertex form: Write the function in the form $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex.
3. Find the axis of symmetry: The axis of symmetry is the vertical line $x=h$.

Q: How do I find the maximum or minimum value of a quadratic function?

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To find the maximum or minimum value of a quadratic function, you can use the following steps:

1. Identify the vertex: Identify the vertex of the parabola, which is the point $(h,k)$.
2. Write the function in vertex form: Write the function in the form $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex.
3. Find the maximum or minimum value: The maximum or minimum value is the point $(h,k)$.

Conclusion

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Quadratic functions are a type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These functions have a parabolic shape and can have various features such as $x$-intercepts, $y$-intercepts, maximum or minimum values, and an axis of symmetry. By understanding the features of a quadratic function and how to find the values of $a$, $b$, and $c$, you can solve problems involving quadratic functions and write them in vertex form.

Additional Resources

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• Quadratic Function Formula: $f(x) = ax^2 + bx + c$
• Vertex Form of a Quadratic Function: $f(x) = a(x-h)^2 + k$
• Axis of Symmetry: $x=h$
• Maximum or Minimum Value: $(h,k)$

Practice Problems

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1. Find the value of $a$, $b$, and $c$ in the quadratic function $f(x) = ax^2 + bx + c$ given that the $x$-intercept is $(8,0)$, the $y$-intercept is $(0,-32)$, and the maximum value is $(6,4)$.
2. Write the quadratic function $f(x) = ax^2 + bx + c$ in vertex form given that the vertex is $(h,k)$.
3. Find the axis of symmetry of the quadratic function $f(x) = ax^2 + bx + c$ given that the vertex is $(h,k)$.
4. Find the maximum or minimum value of the quadratic function $f(x) = ax^2 + bx + c$ given that the vertex is $(h,k)$.