Graphing the Standard Quadratic Function

Oct 13, 2025 by miodeya.com 93 views

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and geometry. A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The standard quadratic function is represented by the equation f(x) = x^2. In this article, we will explore how to graph the standard quadratic function and then use transformations to graph a given function, h(x) = (x - 7)^2 + 3.

Graphing the Standard Quadratic Function

The standard quadratic function is f(x) = x^2. To graph this function, we can start by identifying its key features. The function is a parabola that opens upwards, with its vertex at the origin (0, 0). The parabola is symmetric about the y-axis, and its axis of symmetry is the vertical line x = 0.

To graph the function, we can start by plotting the vertex at (0, 0). Then, we can plot additional points on the graph by substituting different values of x into the equation and calculating the corresponding values of y.

Plotting Points on the Graph

Let's plot some points on the graph by substituting different values of x into the equation f(x) = x^2.

x	f(x) = x^2
-2	4
-1	1
0	0
1	1
2	4

By plotting these points on the graph, we can see that the graph of the standard quadratic function is a parabola that opens upwards.

Transforming the Graph

Now that we have graphed the standard quadratic function, we can use transformations to graph the given function, h(x) = (x - 7)^2 + 3. A transformation is a way of changing the graph of a function by shifting, scaling, or reflecting it.

In this case, we need to perform two transformations:

Horizontal Shift: The function h(x) = (x - 7)^2 + 3 has a horizontal shift of 7 units to the right. This means that the graph of h(x) will be the same as the graph of f(x) = x^2, but shifted 7 units to the right.
Vertical Shift: The function h(x) = (x - 7)^2 + 3 also has a vertical shift of 3 units upwards. This means that the graph of h(x) will be the same as the graph of f(x) = x^2, but shifted 3 units upwards.

Graphing the Transformed Function

To graph the transformed function, we can start by graphing the standard quadratic function and then applying the two transformations.

Step 1: Horizontal Shift

To apply the horizontal shift, we need to shift the graph of f(x) = x^2 7 units to the right. This means that the new x-values will be x - 7, and the new y-values will remain the same.

Step 2: Vertical Shift

To apply the vertical shift, we need to shift the graph of f(x) = x^2 3 units upwards. This means that the new y-values will be y + 3, and the new x-values will remain the same.

Combining the Transformations

To combine the two transformations, we can substitute the new x-values into the equation f(x) = x^2 and then add 3 to the result.

h(x) = (x - 7)^2 + 3

This is the equation of the transformed function.

Graphing the Transformed Function

To graph the transformed function, we can start by graphing the standard quadratic function and then applying the two transformations.

Step 1: Graphing the Standard Quadratic Function

Step 2: Applying the Horizontal Shift

To apply the horizontal shift, we need to shift the graph of f(x) = x^2 7 units to the right. This means that the new x-values will be x - 7, and the new y-values will remain the same.

Step 3: Applying the Vertical Shift

To apply the vertical shift, we need to shift the graph of f(x) = x^2 3 units upwards. This means that the new y-values will be y + 3, and the new x-values will remain the same.

Step 4: Combining the Transformations

To combine the two transformations, we can substitute the new x-values into the equation f(x) = x^2 and then add 3 to the result.

h(x) = (x - 7)^2 + 3

This is the equation of the transformed function.

Conclusion

In this article, we have explored how to graph the standard quadratic function and then use transformations to graph a given function, h(x) = (x - 7)^2 + 3. We have seen how to apply horizontal and vertical shifts to the graph of the standard quadratic function to obtain the graph of the transformed function. By following these steps, we can graph any quadratic function that has been transformed from the standard quadratic function.

Key Takeaways

The standard quadratic function is f(x) = x^2.
The graph of the standard quadratic function is a parabola that opens upwards, with its vertex at the origin (0, 0).
To graph the transformed function, we need to apply two transformations: a horizontal shift of 7 units to the right and a vertical shift of 3 units upwards.
The equation of the transformed function is h(x) = (x - 7)^2 + 3.

Final Thoughts

Q: What is the standard quadratic function?

A: The standard quadratic function is f(x) = x^2. It is a polynomial function of degree two, which means the highest power of the variable (usually x) is two.

Q: What is the graph of the standard quadratic function?

A: The graph of the standard quadratic function is a parabola that opens upwards, with its vertex at the origin (0, 0). The parabola is symmetric about the y-axis, and its axis of symmetry is the vertical line x = 0.

Q: How do I graph the standard quadratic function?

A: To graph the standard quadratic function, you can start by plotting the vertex at (0, 0). Then, you can plot additional points on the graph by substituting different values of x into the equation and calculating the corresponding values of y.

Q: What is a transformation in mathematics?

A: A transformation is a way of changing the graph of a function by shifting, scaling, or reflecting it. In the context of quadratic functions, transformations can be used to shift the graph of the standard quadratic function to obtain the graph of a given function.

Q: What are the two transformations used to graph the given function h(x) = (x - 7)^2 + 3?

A: The two transformations used to graph the given function h(x) = (x - 7)^2 + 3 are:

Horizontal Shift: The function h(x) = (x - 7)^2 + 3 has a horizontal shift of 7 units to the right. This means that the graph of h(x) will be the same as the graph of f(x) = x^2, but shifted 7 units to the right.
Vertical Shift: The function h(x) = (x - 7)^2 + 3 also has a vertical shift of 3 units upwards. This means that the graph of h(x) will be the same as the graph of f(x) = x^2, but shifted 3 units upwards.

Q: How do I apply the horizontal shift to the graph of the standard quadratic function?

A: To apply the horizontal shift, you need to shift the graph of f(x) = x^2 7 units to the right. This means that the new x-values will be x - 7, and the new y-values will remain the same.

Q: How do I apply the vertical shift to the graph of the standard quadratic function?

A: To apply the vertical shift, you need to shift the graph of f(x) = x^2 3 units upwards. This means that the new y-values will be y + 3, and the new x-values will remain the same.

Q: What is the equation of the transformed function h(x) = (x - 7)^2 + 3?

A: The equation of the transformed function h(x) = (x - 7)^2 + 3 is obtained by substituting the new x-values into the equation f(x) = x^2 and then adding 3 to the result.

Q: How do I graph the transformed function h(x) = (x - 7)^2 + 3?

A: To graph the transformed function h(x) = (x - 7)^2 + 3, you can start by graphing the standard quadratic function and then applying the two transformations: a horizontal shift of 7 units to the right and a vertical shift of 3 units upwards.

Q: What are some key takeaways from this article?

A: Some key takeaways from this article are:

The standard quadratic function is f(x) = x^2.
The graph of the standard quadratic function is a parabola that opens upwards, with its vertex at the origin (0, 0).
To graph the transformed function, you need to apply two transformations: a horizontal shift of 7 units to the right and a vertical shift of 3 units upwards.
The equation of the transformed function is h(x) = (x - 7)^2 + 3.

Q: What are some final thoughts on transforming quadratic functions?

A: Graphing quadratic functions and applying transformations is an essential skill in mathematics. By understanding how to graph the standard quadratic function and then use transformations to graph a given function, you can solve a wide range of problems in algebra and geometry.