Select The Correct Answer From Each Drop-down Menu.Identify The Key Features Of These Functions:${ \begin{array}{l} f(x)=3(x+5)^2-10 \ g(x)=-4x^2+3x-1 \ h(x)=-0.25(x+3)(x-1) \end{array} }$- The Vertex Of { F $}$ Is At

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their key features is essential for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will explore the key features of three quadratic functions: $f(x)=3(x+5)^2-10$, $g(x)=-4x^2+3x-1$, and $h(x)=-0.25(x+3)(x-1)$. We will identify the vertex, axis of symmetry, and other important characteristics of each function.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. To identify the vertex of each function, we need to rewrite them in vertex form.

Function $f(x)=3(x+5)^2-10$

To rewrite $f(x)$ in vertex form, we need to expand the squared term:

$$f(x)=3(x^2+10x+25)-10$$

$$f(x)=3x^2+30x+75-10$$

$$f(x)=3x^2+30x+65$$

Now, we can identify the vertex by comparing the expanded form with the vertex form:

$$f(x)=3(x-(-5))^2+65$$

The vertex of $f(x)$ is at $(-5,65)$.

Function $g(x)=-4x^2+3x-1$

To rewrite $g(x)$ in vertex form, we need to complete the square:

$$g(x)=-4x^2+3x-1$$

$$g(x)=-4(x^2-\frac{3}{4}x)+(-1)$$

$$g(x)=-4(x^2-\frac{3}{4}x+\frac{9}{16})+(-1)+4(\frac{9}{16})$$

$$g(x)=-4(x-\frac{3}{8})^2+\frac{35}{16}$$

The vertex of $g(x)$ is at $(\frac{3}{8},\frac{35}{16})$.

Function $h(x)=-0.25(x+3)(x-1)$

To rewrite $h(x)$ in vertex form, we need to expand the product:

$$h(x)=-0.25(x^2+2x-3)$$

$$h(x)=-0.25x^2-0.5x+0.75$$

Now, we can identify the vertex by comparing the expanded form with the vertex form:

$$h(x)=-0.25(x+3)(x-1)$$

The vertex of $h(x)$ is at $(-3,0.75)$.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is given by $x=h$, where $(h,k)$ is the vertex of the parabola.

Function $f(x)=3(x+5)^2-10$

The axis of symmetry of $f(x)$ is given by $x=-5$.

Function $g(x)=-4x^2+3x-1$

The axis of symmetry of $g(x)$ is given by $x=\frac{3}{8}$.

Function $h(x)=-0.25(x+3)(x-1)$

The axis of symmetry of $h(x)$ is given by $x=-1.5$.

Domain and Range

The domain of a quadratic function is the set of all possible input values for which the function is defined. The range of a quadratic function is the set of all possible output values.

Function $f(x)=3(x+5)^2-10$

The domain of $f(x)$ is all real numbers, and the range is all real numbers greater than or equal to $-10$.

Function $g(x)=-4x^2+3x-1$

The domain of $g(x)$ is all real numbers, and the range is all real numbers less than or equal to $-1$.

Function $h(x)=-0.25(x+3)(x-1)$

The domain of $h(x)$ is all real numbers, and the range is all real numbers greater than or equal to $-0.75$.

Conclusion

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic functions. We will cover topics such as vertex form, axis of symmetry, domain, and range.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola.

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

A: To find the vertex of a quadratic function, you need to rewrite it in vertex form. This can be done by completing the square or using the formula $h=-\frac{b}{2a}$.

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is given by $x=h$, where $(h,k)$ is the vertex of the parabola.

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

A: To find the axis of symmetry of a quadratic function, you need to find the vertex of the function and then use the formula $x=h$.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values for which the function is defined. For a quadratic function, the domain is all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values. The range of a quadratic function depends on the vertex and the coefficient of the squared term.

Q: How do I determine the range of a quadratic function?

A: To determine the range of a quadratic function, you need to find the vertex and the coefficient of the squared term. Then, you can use the formula $k=a(x-h)^2$ to find the minimum or maximum value of the function.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. In this case, the parabola will open downward, and the vertex will be the maximum point.

Q: Can a quadratic function have a zero leading coefficient?

A: No, a quadratic function cannot have a zero leading coefficient. If the leading coefficient is zero, then the function is not quadratic.

Q: Can a quadratic function have a negative vertex?

A: Yes, a quadratic function can have a negative vertex. In this case, the parabola will open downward, and the vertex will be the maximum point.

Q: Can a quadratic function have a positive vertex?

A: Yes, a quadratic function can have a positive vertex. In this case, the parabola will open upward, and the vertex will be the minimum point.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have covered topics such as vertex form, axis of symmetry, domain, and range. Understanding these concepts is essential for solving various problems in algebra, calculus, and other branches of mathematics.

Additional Resources

For more information on quadratic functions, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Wolfram MathWorld: Quadratic Functions

We hope this article has been helpful in answering your questions about quadratic functions. If you have any further questions, please don't hesitate to ask.