The Graph Of Which Function Will Have A Maximum And A $y$-intercept Of 4?A. F ( X ) = 4 X 2 + 6 X − 1 F(x) = 4x^2 + 6x - 1 F ( X ) = 4 X 2 + 6 X − 1 B. F ( X ) = − 4 X 2 + 8 X + 5 F(x) = -4x^2 + 8x + 5 F ( X ) = − 4 X 2 + 8 X + 5 C. F ( X ) = − X 2 + 2 X + 4 F(x) = -x^2 + 2x + 4 F ( X ) = − X 2 + 2 X + 4 D. F ( X ) = X 2 + 4 X − 4 F(x) = X^2 + 4x - 4 F ( X ) = X 2 + 4 X − 4

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The Graph of Which Function Will Have a Maximum and a $y$-intercept of 4?

Understanding the Problem

To solve this problem, we need to analyze each given function and determine which one has a maximum and a $y$-intercept of 4. The $y$-intercept is the point at which the graph of the function crosses the $y$-axis, and it is found by setting $x = 0$ in the function.

Analyzing the Functions

A. f(x)=4x2+6x−1f(x) = 4x^2 + 6x - 1

This is a quadratic function with a positive leading coefficient, which means that its graph will be a parabola that opens upward. To find the $y$-intercept, we set $x = 0$:

f(0)=4(0)2+6(0)−1=−1f(0) = 4(0)^2 + 6(0) - 1 = -1

Since the $y$-intercept is not 4, this function does not meet the requirements.

B. f(x)=−4x2+8x+5f(x) = -4x^2 + 8x + 5

This is also a quadratic function, but with a negative leading coefficient, which means that its graph will be a parabola that opens downward. To find the $y$-intercept, we set $x = 0$:

f(0)=−4(0)2+8(0)+5=5f(0) = -4(0)^2 + 8(0) + 5 = 5

Since the $y$-intercept is not 4, this function does not meet the requirements.

C. f(x)=−x2+2x+4f(x) = -x^2 + 2x + 4

This is another quadratic function with a negative leading coefficient, which means that its graph will be a parabola that opens downward. To find the $y$-intercept, we set $x = 0$:

f(0)=−(0)2+2(0)+4=4f(0) = -(0)^2 + 2(0) + 4 = 4

This function meets the requirement of having a $y$-intercept of 4.

D. f(x)=x2+4x−4f(x) = x^2 + 4x - 4

This is a quadratic function with a positive leading coefficient, which means that its graph will be a parabola that opens upward. To find the $y$-intercept, we set $x = 0$:

f(0)=(0)2+4(0)−4=−4f(0) = (0)^2 + 4(0) - 4 = -4

Since the $y$-intercept is not 4, this function does not meet the requirements.

Conclusion

Based on the analysis of each function, the graph of the function f(x)=−x2+2x+4f(x) = -x^2 + 2x + 4 will have a maximum and a $y$-intercept of 4.

Why is this function the correct answer?

This function is the correct answer because it meets both requirements: it has a $y$-intercept of 4, and its graph is a parabola that opens downward, which means that it has a maximum. The other functions either do not have a $y$-intercept of 4 or do not have a maximum.

What is the significance of this problem?

This problem is significant because it requires the analysis of quadratic functions and their graphs. Understanding the properties of quadratic functions is essential in mathematics and has many real-world applications, such as modeling population growth, projectile motion, and electrical circuits.

How can this problem be extended?

This problem can be extended by asking students to find the maximum value of the function f(x)=−x2+2x+4f(x) = -x^2 + 2x + 4 or to graph the function and identify its maximum point. Additionally, students can be asked to analyze other quadratic functions and determine which ones meet the requirements of having a maximum and a $y$-intercept of 4.

What are the key concepts in this problem?

The key concepts in this problem are:

  • Quadratic functions

  • y$-intercept

  • Maximum value
  • Graphing quadratic functions
  • Analyzing the properties of quadratic functions

What are the skills required to solve this problem?

The skills required to solve this problem are:

  • Analyzing quadratic functions
  • Finding the $y$-intercept of a quadratic function
  • Identifying the maximum value of a quadratic function
  • Graphing quadratic functions
  • Analyzing the properties of quadratic functions
    Q&A: The Graph of Which Function Will Have a Maximum and a $y$-intercept of 4?

Frequently Asked Questions

Q: What is the main goal of this problem?

A: The main goal of this problem is to determine which quadratic function has a maximum and a $y$-intercept of 4.

Q: What is the significance of the $y$-intercept in this problem?

A: The $y$-intercept is the point at which the graph of the function crosses the $y$-axis, and it is found by setting $x = 0$ in the function. In this problem, we are looking for a function that has a $y$-intercept of 4.

Q: What is the difference between a maximum and a minimum value in a quadratic function?

A: A maximum value is the highest point on the graph of a quadratic function, while a minimum value is the lowest point on the graph. In this problem, we are looking for a function that has a maximum value.

Q: How can I determine if a quadratic function has a maximum or a minimum value?

A: To determine if a quadratic function has a maximum or a minimum value, you can look at the leading coefficient of the function. If the leading coefficient is positive, the function has a minimum value. If the leading coefficient is negative, the function has a maximum value.

Q: What is the relationship between the leading coefficient and the direction of the parabola?

A: The leading coefficient determines the direction of the parabola. If the leading coefficient is positive, the parabola opens upward. If the leading coefficient is negative, the parabola opens downward.

Q: How can I find the $y$-intercept of a quadratic function?

A: To find the $y$-intercept of a quadratic function, you can set $x = 0$ in the function and solve for $y$.

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

A: The vertex of a quadratic function is the point at which the graph of the function changes direction. It is also the maximum or minimum point on the graph.

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

A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, such as modeling population growth, projectile motion, and electrical circuits.

Q: How can I extend this problem to make it more challenging?

A: You can extend this problem by asking students to find the maximum value of the function $f(x) = -x^2 + 2x + 4$ or to graph the function and identify its maximum point. Additionally, you can ask students to analyze other quadratic functions and determine which ones meet the requirements of having a maximum and a $y$-intercept of 4.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

  • Not setting $x = 0$ to find the $y$-intercept
  • Not identifying the leading coefficient and its effect on the direction of the parabola
  • Not using the formula $x = -\frac{b}{2a}$ to find the vertex of the quadratic function

Q: How can I use technology to help solve this problem?

A: You can use technology such as graphing calculators or computer software to help solve this problem. These tools can be used to graph the functions and identify the $y$-intercept and vertex of the quadratic functions.