The Graph Of The Function F ( X ) = ( X + 6 ) ( X + 2 F(x)=(x+6)(x+2 F ( X ) = ( X + 6 ) ( X + 2 ] Is Shown. Which Statements Describe The Graph? Choose Three Correct Answers.A. The Axis Of Symmetry Is X = − 4 X=-4 X = − 4 .B. The Vertex Is The Maximum Value.C. The Domain Is All Real Numbers.

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Understanding the Graph of a Quadratic Function

The graph of the function $f(x)=(x+6)(x+2)$ is a quadratic function, which is a polynomial function of degree two. The general form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. In this case, the function can be rewritten as $f(x)=x^2+8x+12$.

Identifying the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The vertex is the point on the graph where the parabola changes direction. To find the axis of symmetry, we need to find the x-coordinate of the vertex. The x-coordinate of the vertex can be found using the formula $x=-\frac{b}{2a}$.

In this case, $a=1$ and $b=8$. Plugging these values into the formula, we get $x=-\frac{8}{2(1)}=-4$. Therefore, the axis of symmetry is $x=-4$.

Determining the Vertex

The vertex of a quadratic function is the maximum or minimum value of the function. In this case, the vertex is the maximum value of the function. To find the y-coordinate of the vertex, we need to plug the x-coordinate of the vertex into the function.

The x-coordinate of the vertex is $x=-4$. Plugging this value into the function, we get $f(-4)=(-4+6)(-4+2)=2(-2)=-4$. Therefore, the vertex is the point $(-4,-4)$.

Identifying the Domain

The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers. Therefore, the domain is all real numbers.

Choosing the Correct Statements

Based on the analysis above, the correct statements that describe the graph of the function $f(x)=(x+6)(x+2)$ are:

• A. The axis of symmetry is $x=-4$.
• B. The vertex is the maximum value.
• C. The domain is all real numbers.

The other statements are incorrect. The vertex is not the minimum value, and the domain is not a specific interval.

Conclusion

In conclusion, the graph of the function $f(x)=(x+6)(x+2)$ is a quadratic function with an axis of symmetry at $x=-4$, a vertex at $(-4,-4)$, and a domain of all real numbers. The correct statements that describe the graph are A, B, and C.

Final Answer

The final answer is:

• A. The axis of symmetry is $x=-4$.
• B. The vertex is the maximum value.
• C. The domain is all real numbers.
  

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Understanding the Graph of a Quadratic Function

The graph of the function $f(x)=(x+6)(x+2)$ is a quadratic function, which is a polynomial function of degree two. The general form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. In this case, the function can be rewritten as $f(x)=x^2+8x+12$.

Q&A: The Graph of the Function $f(x)=(x+6)(x+2)$

Q: What is the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$?

A: The axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$ is a vertical line that passes through the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $x=-\frac{b}{2a}$. In this case, $a=1$ and $b=8$. Plugging these values into the formula, we get $x=-\frac{8}{2(1)}=-4$. Therefore, the axis of symmetry is $x=-4$.

Q: What is the vertex of the graph of the function $f(x)=(x+6)(x+2)$?

A: The vertex of the graph of the function $f(x)=(x+6)(x+2)$ is the point on the graph where the parabola changes direction. To find the y-coordinate of the vertex, we need to plug the x-coordinate of the vertex into the function. The x-coordinate of the vertex is $x=-4$. Plugging this value into the function, we get $f(-4)=(-4+6)(-4+2)=2(-2)=-4$. Therefore, the vertex is the point $(-4,-4)$.

Q: What is the domain of the graph of the function $f(x)=(x+6)(x+2)$?

A: The domain of the graph of the function $f(x)=(x+6)(x+2)$ is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers. Therefore, the domain is all real numbers.

Q: What is the maximum value of the graph of the function $f(x)=(x+6)(x+2)$?

A: The maximum value of the graph of the function $f(x)=(x+6)(x+2)$ is the y-coordinate of the vertex. The y-coordinate of the vertex is $-4$. Therefore, the maximum value of the graph is $-4$.

Q: What is the minimum value of the graph of the function $f(x)=(x+6)(x+2)$?

A: The minimum value of the graph of the function $f(x)=(x+6)(x+2)$ is not defined, as the graph is a parabola that opens upwards.

Q: What is the x-intercept of the graph of the function $f(x)=(x+6)(x+2)$?

A: The x-intercept of the graph of the function $f(x)=(x+6)(x+2)$ is the point where the graph crosses the x-axis. To find the x-intercept, we need to set the function equal to zero and solve for x. The function can be rewritten as $f(x)=x^2+8x+12$. Setting this equal to zero, we get $x^2+8x+12=0$. Factoring the quadratic, we get $(x+6)(x+2)=0$. Solving for x, we get $x=-6$ or $x=-2$. Therefore, the x-intercepts are $(-6,0)$ and $(-2,0)$.

Q: What is the y-intercept of the graph of the function $f(x)=(x+6)(x+2)$?

A: The y-intercept of the graph of the function $f(x)=(x+6)(x+2)$ is the point where the graph crosses the y-axis. To find the y-intercept, we need to plug in x=0 into the function. The function can be rewritten as $f(x)=x^2+8x+12$. Plugging in x=0, we get $f(0)=0^2+8(0)+12=12$. Therefore, the y-intercept is $(0,12)$.

Q: What is the vertex form of the graph of the function $f(x)=(x+6)(x+2)$?

A: The vertex form of the graph of the function $f(x)=(x+6)(x+2)$ is $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(-4,-4)$. Therefore, the vertex form of the graph is $f(x)=1(x-(-4))^2-4$.

Q: What is the standard form of the graph of the function $f(x)=(x+6)(x+2)$?

A: The standard form of the graph of the function $f(x)=(x+6)(x+2)$ is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. In this case, the function can be rewritten as $f(x)=x^2+8x+12$.

Q: What is the equation of the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$ is $x=-4$.

Q: What is the equation of the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$ is $y=-4$.

Q: What is the equation of the line that is parallel to the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that is parallel to the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$ is $x=-4$.

Q: What is the equation of the line that is perpendicular to the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that is perpendicular to the axis of symmetry of the graph of the function $f(x)=(x+6)(x+2)$ is $y=-4$.

Q: What is the equation of the line that passes through the x-intercept of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that passes through the x-intercept of the graph of the function $f(x)=(x+6)(x+2)$ is $y=0$.

Q: What is the equation of the line that passes through the y-intercept of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that passes through the y-intercept of the graph of the function $f(x)=(x+6)(x+2)$ is $x=0$.

Q: What is the equation of the line that is parallel to the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that is parallel to the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$ is $y=-4$.

Q: What is the equation of the line that is perpendicular to the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$?

A: The equation of the line that is perpendicular to the line that passes through the vertex of the graph of the function $f(x)=(x+6)(x+2)$ is $x=-4$.

Q: What is the equation of the line that passes through the point $(-4,-4)$?

A: The equation of the line that passes through the point $(-4,-4)$ is $y=-4$.

Q: What is the equation of the line that passes through the point $(-6,0)$?

A: The equation of the line that passes through the point $(-6,0)$ is $y=0$.

Q: What is the equation of the line that passes through the point $(-2,0)$?

A: The equation of the line that passes through the point $(-2,0)$ is $y=0$.

Q: What is the equation of the line that passes through the point $(0,12)$?

A: The equation of the line that passes through the point $(0,12)$ is $x=0$.

Q: What is the equation of the line that is parallel to the line that passes through the point $(-4,-4)$?

A: The equation of the line that is parallel to the line that passes through the point $(-4,-4)$ is $y=-4$.