Salina Finds The Range Of The Function $y = -4x^2 + 8x + 7$ By Calculating The Vertex. Her Work Is Shown Below. Determine Her Error And Find The Correct Range Of The Function.Given:Vertex: (1, 11)Range: $y \geq 11$(Note: The Error

Introduction

In mathematics, finding the range of a quadratic function is a crucial step in understanding the behavior of the function. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will discuss how to find the range of a quadratic function by calculating the vertex, and we will also identify the error made by Salina in her work.

The Vertex Formula

The vertex of a quadratic function is the maximum or minimum point of the function. The vertex formula is given by:

$$x = -\frac{b}{2a}$$

This formula gives the x-coordinate of the vertex. To find the y-coordinate of the vertex, we substitute the x-coordinate into the function.

Salina's Work

Salina was given the function $y = -4x^2 + 8x + 7$ and was asked to find the range of the function by calculating the vertex. Her work is shown below:

1. Find the x-coordinate of the vertex using the vertex formula:

$$x = -\frac{b}{2a} = -\frac{8}{2(-4)} = 1$$

2. Substitute the x-coordinate into the function to find the y-coordinate of the vertex:

$$y = -4(1)^2 + 8(1) + 7 = -4 + 8 + 7 = 11$$

3. Write the range of the function:

$$y \geq 11$$

The Error

Salina's work looks correct, but there is an error in her conclusion. The error is that she wrote the range of the function as $y \geq 11$, which means that the function has a minimum value of 11. However, the function $y = -4x^2 + 8x + 7$ is a quadratic function with a negative leading coefficient, which means that the function has a maximum value, not a minimum value.

The Correct Range

To find the correct range of the function, we need to analyze the function. Since the function is a quadratic function with a negative leading coefficient, the function has a maximum value. The maximum value of the function occurs at the vertex, which is $(1, 11)$. Since the function is a quadratic function, the graph of the function is a parabola that opens downward. This means that the function has a maximum value of 11, and the range of the function is $y \leq 11$.

Conclusion

In conclusion, Salina's work was correct in finding the vertex of the function, but her conclusion was incorrect. The correct range of the function is $y \leq 11$, not $y \geq 11$. This is because the function is a quadratic function with a negative leading coefficient, which means that the function has a maximum value, not a minimum value.

The Importance of Understanding the Vertex

Understanding the vertex of a quadratic function is crucial in finding the range of the function. The vertex formula is a powerful tool that helps us find the x-coordinate of the vertex. By substituting the x-coordinate into the function, we can find the y-coordinate of the vertex. The vertex of a quadratic function is the maximum or minimum point of the function, and it helps us understand the behavior of the function.

The Importance of Analyzing the Function

Analyzing the function is also crucial in finding the range of the function. By analyzing the function, we can determine whether the function has a maximum or minimum value. If the function has a negative leading coefficient, the function has a maximum value. If the function has a positive leading coefficient, the function has a minimum value.

The Importance of Writing the Correct Range

Writing the correct range of the function is also crucial. The range of the function should be written in the correct format, which is $y \leq k$ or $y \geq k$, where $k$ is the maximum or minimum value of the function.

The Final Answer

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Introduction

In our previous article, we discussed how to find the range of a quadratic function by calculating the vertex. We also identified the error made by Salina in her work and found the correct range of the function. In this article, we will answer some frequently asked questions about finding the range of a quadratic function.

Q: What is the vertex formula?

A: The vertex formula is given by:

$$x = -\frac{b}{2a}$$

This formula gives the x-coordinate of the vertex. To find the y-coordinate of the vertex, we substitute the x-coordinate into the function.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, we substitute the x-coordinate into the function. For example, if we have the function $y = -4x^2 + 8x + 7$ and we find that the x-coordinate of the vertex is 1, we substitute x = 1 into the function to get:

$$y = -4(1)^2 + 8(1) + 7 = -4 + 8 + 7 = 11$$

Q: What is the range of a quadratic function with a negative leading coefficient?

A: The range of a quadratic function with a negative leading coefficient is $y \leq k$, where $k$ is the maximum value of the function. This is because the function has a maximum value, and the graph of the function is a parabola that opens downward.

Q: What is the range of a quadratic function with a positive leading coefficient?

A: The range of a quadratic function with a positive leading coefficient is $y \geq k$, where $k$ is the minimum value of the function. This is because the function has a minimum value, and the graph of the function is a parabola that opens upward.

Q: How do I determine whether a quadratic function has a maximum or minimum value?

A: To determine whether a quadratic function has a maximum or minimum value, we look at the leading coefficient. If the leading coefficient is negative, the function has a maximum value. If the leading coefficient is positive, the function has a minimum value.

Q: What is the importance of understanding the vertex of a quadratic function?

A: Understanding the vertex of a quadratic function is crucial in finding the range of the function. The vertex formula is a powerful tool that helps us find the x-coordinate of the vertex. By substituting the x-coordinate into the function, we can find the y-coordinate of the vertex. The vertex of a quadratic function is the maximum or minimum point of the function, and it helps us understand the behavior of the function.

Q: What is the importance of analyzing the function?

A: Analyzing the function is also crucial in finding the range of the function. By analyzing the function, we can determine whether the function has a maximum or minimum value. If the function has a negative leading coefficient, the function has a maximum value. If the function has a positive leading coefficient, the function has a minimum value.

Q: What is the importance of writing the correct range of the function?

A: Writing the correct range of the function is also crucial. The range of the function should be written in the correct format, which is $y \leq k$ or $y \geq k$, where $k$ is the maximum or minimum value of the function.

Conclusion

In conclusion, finding the range of a quadratic function is a crucial step in understanding the behavior of the function. By using the vertex formula and analyzing the function, we can determine whether the function has a maximum or minimum value. We can then write the correct range of the function in the correct format.