Use The Given Function To Find The Information. F ( X ) = − ( X − 2 ) 2 + 9 F(x) = -(x - 2)^2 + 9 F ( X ) = − ( X − 2 ) 2 + 9 A. Intercepts

Understanding the Function

The given function is $f(x) = -(x - 2)^2 + 9$. This is a quadratic function in the form of $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(2, 9)$.

Finding Intercepts

To find the x-intercepts, we need to set $f(x) = 0$ and solve for $x$. This means that the function will intersect the x-axis at the points where the graph touches the x-axis.

Finding the x-Intercepts

To find the x-intercepts, we set $f(x) = 0$ and solve for $x$:

$$-(x - 2)^2 + 9 = 0$$

We can start by isolating the squared term:

$$(x - 2)^2 = 9$$

Taking the square root of both sides, we get:

$$x - 2 = \pm 3$$

Solving for $x$, we get:

$$x = 2 \pm 3$$

So, the x-intercepts are $x = 5$ and $x = -1$.

Finding the y-Intercept

To find the y-intercept, we need to find the value of $f(x)$ when $x = 0$. This is the point where the graph touches the y-axis.

We can plug in $x = 0$ into the function:

$$f(0) = -(0 - 2)^2 + 9$$

Simplifying, we get:

$$f(0) = -(-2)^2 + 9$$

$$f(0) = -4 + 9$$

$$f(0) = 5$$

So, the y-intercept is $(0, 5)$.

Finding the Vertex

The vertex of the parabola is the point where the function changes from decreasing to increasing or vice versa. In this case, the vertex is $(2, 9)$.

Finding the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is $x = 2$.

Finding the Maximum or Minimum Value

The maximum or minimum value of the function is the value of the function at the vertex. In this case, the maximum value is $9$.

Graphing the Function

To graph the function, we can use the information we have found. We can plot the x-intercepts, y-intercept, vertex, and axis of symmetry on a coordinate plane.

Conclusion

In this article, we used the given function to find the x-intercepts, y-intercept, vertex, axis of symmetry, and maximum or minimum value. We also graphed the function using the information we found. This demonstrates how to use a quadratic function to find important information about the graph of the function.

Final Answer

The final answer is:

• x-intercepts: $x = 5$ and $x = -1$
• y-intercept: $(0, 5)$
• vertex: $(2, 9)$
• axis of symmetry: $x = 2$
• maximum or minimum value: $9$
  

Understanding the Function

The given function is $f(x) = -(x - 2)^2 + 9$. This is a quadratic function in the form of $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(2, 9)$.

Q&A

Q: What is the x-intercept of the function?

A: To find the x-intercept, we need to set $f(x) = 0$ and solve for $x$. This means that the function will intersect the x-axis at the points where the graph touches the x-axis.

Q: How do I find the x-intercepts of the function?

A: To find the x-intercepts, we set $f(x) = 0$ and solve for $x$. We can start by isolating the squared term:

$$(x - 2)^2 = 9$$

Taking the square root of both sides, we get:

$$x - 2 = \pm 3$$

Solving for $x$, we get:

$$x = 2 \pm 3$$

So, the x-intercepts are $x = 5$ and $x = -1$.

Q: What is the y-intercept of the function?

A: To find the y-intercept, we need to find the value of $f(x)$ when $x = 0$. This is the point where the graph touches the y-axis.

We can plug in $x = 0$ into the function:

$$f(0) = -(0 - 2)^2 + 9$$

Simplifying, we get:

$$f(0) = -(-2)^2 + 9$$

$$f(0) = -4 + 9$$

$$f(0) = 5$$

So, the y-intercept is $(0, 5)$.

Q: What is the vertex of the function?

A: The vertex of the parabola is the point where the function changes from decreasing to increasing or vice versa. In this case, the vertex is $(2, 9)$.

Q: What is the axis of symmetry of the function?

A: The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is $x = 2$.

Q: What is the maximum or minimum value of the function?

A: The maximum or minimum value of the function is the value of the function at the vertex. In this case, the maximum value is $9$.

Q: How do I graph the function?

A: To graph the function, we can use the information we have found. We can plot the x-intercepts, y-intercept, vertex, and axis of symmetry on a coordinate plane.

Q: What is the significance of the given function?

A: The given function is a quadratic function, which is a polynomial function of degree two. It has a parabolic shape and can be used to model a wide range of real-world phenomena, such as the trajectory of a projectile or the growth of a population.

Conclusion

In this article, we answered some common questions about the given quadratic function. We discussed how to find the x-intercepts, y-intercept, vertex, axis of symmetry, and maximum or minimum value of the function. We also graphed the function using the information we found. This demonstrates how to use a quadratic function to find important information about the graph of the function.

Final Answer

The final answer is:

• x-intercepts: $x = 5$ and $x = -1$
• y-intercept: $(0, 5)$
• vertex: $(2, 9)$
• axis of symmetry: $x = 2$
• maximum or minimum value: $9$