Understanding the Features of the Graph

Oct 13, 2025 by windailys.com 64 views

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and are used to model various real-world phenomena. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The graph of a quadratic function is a parabola, and it can be either upward-facing or downward-facing. In this article, we will discuss how to select the quadratic function with a graph that has specific features.

Understanding the Features of the Graph

The given features of the graph are:

x-intercept at (8,0): This means that the graph touches the x-axis at the point (8,0). In other words, when y = 0, x = 8.
y-intercept at (0,-32): This means that the graph touches the y-axis at the point (0,-32). In other words, when x = 0, y = -32.
Maximum value at (6,4): This means that the graph reaches its maximum value at the point (6,4). In other words, when x = 6, y = 4.
Axis of symmetry at x = 6: This means that the graph is symmetric about the line x = 6. In other words, if we reflect the graph about the line x = 6, we will get the same graph.

The General Form of a Quadratic Function

The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where a, b, and c are constants.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is:

f(x) = a(x - h)^2 + k

where (h,k) is the vertex of the parabola.

The Axis of Symmetry

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. In other words, it is the line x = h.

The x-Intercept and y-Intercept

The x-intercept of a parabola is the point where the graph touches the x-axis. In other words, it is the point where y = 0.

The y-intercept of a parabola is the point where the graph touches the y-axis. In other words, it is the point where x = 0.

The Maximum Value

The maximum value of a parabola is the highest point on the graph. In other words, it is the point where the graph reaches its highest value.

Selecting the Quadratic Function

Based on the given features of the graph, we can select the quadratic function as follows:

x-intercept at (8,0): This means that the graph touches the x-axis at the point (8,0). In other words, when y = 0, x = 8. This implies that the quadratic function has a root at x = 8.
y-intercept at (0,-32): This means that the graph touches the y-axis at the point (0,-32). In other words, when x = 0, y = -32. This implies that the quadratic function has a root at y = -32.
Maximum value at (6,4): This means that the graph reaches its maximum value at the point (6,4). In other words, when x = 6, y = 4. This implies that the quadratic function has a vertex at (6,4).
Axis of symmetry at x = 6: This means that the graph is symmetric about the line x = 6. In other words, if we reflect the graph about the line x = 6, we will get the same graph. This implies that the axis of symmetry of the quadratic function is x = 6.

The Quadratic Function

Based on the above analysis, we can select the quadratic function as:

f(x) = a(x - 6)^2 + 4

where a is a constant.

Finding the Value of a

To find the value of a, we can use the fact that the y-intercept of the graph is (0,-32). This means that when x = 0, y = -32. Substituting these values into the quadratic function, we get:

-32 = a(0 - 6)^2 + 4

Simplifying this equation, we get:

-32 = 36a + 4

Subtracting 4 from both sides, we get:

-36 = 36a

Dividing both sides by 36, we get:

a = -1

The Final Quadratic Function

Substituting the value of a into the quadratic function, we get:

f(x) = -(x - 6)^2 + 4

This is the quadratic function with a graph that has the given features.

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: 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 f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can be either upward-facing or downward-facing.

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

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex of the parabola. In other words, it is the line x = h, where (h,k) is the vertex of the parabola.

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

A: The x-intercept of a quadratic function is the point where the graph touches the x-axis. In other words, it is the point where y = 0.

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

A: The y-intercept of a quadratic function is the point where the graph touches the y-axis. In other words, it is the point where x = 0.

Q: What is the maximum value of a quadratic function?

A: The maximum value of a quadratic function is the highest point on the graph. In other words, it is the point where the graph reaches its highest value.

Q: How do I find the quadratic function with a given graph?

A: To find the quadratic function with a given graph, you need to analyze the features of the graph and use them to select the quadratic function. The features of the graph include the x-intercept, y-intercept, axis of symmetry, and maximum value.

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

A: To find the value of a in a quadratic function, you can use the fact that the y-intercept of the graph is (0,-32). This means that when x = 0, y = -32. Substituting these values into the quadratic function, you can solve for a.

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

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

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

A: The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: Can a quadratic function have more than one x-intercept?

A: No, a quadratic function cannot have more than one x-intercept. The x-intercept is the point where the graph touches the x-axis, and there is only one point where the graph touches the x-axis.

Q: Can a quadratic function have more than one y-intercept?

A: No, a quadratic function cannot have more than one y-intercept. The y-intercept is the point where the graph touches the y-axis, and there is only one point where the graph touches the y-axis.

Q: Can a quadratic function have a maximum value at more than one point?

A: No, a quadratic function cannot have a maximum value at more than one point. The maximum value is the highest point on the graph, and there is only one point where the graph reaches its highest value.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We discussed the definition of a quadratic function, the graph of a quadratic function, the axis of symmetry, the x-intercept, the y-intercept, the maximum value, and how to find the quadratic function with a given graph. We also discussed the vertex form and the general form of a quadratic function.