To Graph The Function $g(x)=(x-5)^2-9$, Shift The Graph Of $f(x)=x^2$ Right 5 Units And Down 9 Units.

Mar 11, 2025 by ADMIN 106 views

To Graph the Function $g(x)=(x-5)^2-9$, Shift the Graph of $f(x)=x^2$ Right 5 Units and Down 9 Units

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Introduction

Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. It involves visualizing the relationship between the input and output values of a function. In this article, we will discuss how to graph the function $g(x)=(x-5)^2-9$ by shifting the graph of $f(x)=x^2$ right 5 units and down 9 units.

Understanding the Parent Function

The parent function is the basic function from which other functions are derived. In this case, the parent function is $f(x)=x^2$ . This function is a quadratic function that represents a parabola opening upwards with its vertex at the origin (0, 0).

Characteristics of the Parent Function

* Vertex: (0, 0)

* Axis of Symmetry: x-axis

* Direction of Opening: Upwards

Shifting the Graph of the Parent Function

To graph the function $g(x)=(x-5)^2-9$ , we need to shift the graph of $f(x)=x^2$ right 5 units and down 9 units. This means that we need to adjust the x-coordinate and the y-coordinate of the graph of $f(x)=x^2$ to obtain the graph of $g(x)=(x-5)^2-9$ .

Horizontal Shift

* Right 5 units: To shift the graph of $f(x)=x^2$ right 5 units, we need to replace x with (x - 5) in the equation of $f(x)=x^2$ . This gives us the equation $f(x-5)=(x-5)^2$ .

Vertical Shift

* Down 9 units: To shift the graph of $f(x)=x^2$ down 9 units, we need to subtract 9 from the equation of $f(x)=x^2$ . This gives us the equation $g(x)=(x-5)^2-9$ .

Graphing the Function $g(x)=(x-5)^2-9$

Now that we have the equation of the function $g(x)=(x-5)^2-9$ , we can graph it. To graph the function, we need to plot several points on the graph and then connect them to form a smooth curve.

Graphing the Function

* Plotting Points: To plot points on the graph, we need to substitute different values of x into the equation of $g(x)=(x-5)^2-9$ and calculate the corresponding values of y.

Example

* Plotting the Point (0, -9): To plot the point (0, -9), we need to substitute x = 0 into the equation of $g(x)=(x-5)^2-9$ . This gives us the equation $g(0)=(0-5)^2-9=-9$ . Therefore, the point (0, -9) lies on the graph of $g(x)=(x-5)^2-9$ .

Conclusion

In conclusion, graphing the function $g(x)=(x-5)^2-9$ involves shifting the graph of $f(x)=x^2$ right 5 units and down 9 units. This can be achieved by replacing x with (x - 5) in the equation of $f(x)=x^2$ and then subtracting 9 from the resulting equation. By plotting several points on the graph and connecting them to form a smooth curve, we can visualize the graph of the function $g(x)=(x-5)^2-9$ .

Applications of Graphing Functions

Graphing functions has numerous applications in mathematics, science, and engineering. Some of the applications of graphing functions include:

* Optimization Problems: Graphing functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.

* Modeling Real-World Situations: Graphing functions can be used to model real-world situations, such as the motion of an object or the growth of a population.

* Analyzing Data: Graphing functions can be used to analyze data, such as the relationship between two variables.

Final Thoughts

Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. By understanding how to graph functions, we can visualize the relationship between the input and output values of a function. This can be achieved by shifting the graph of a parent function and plotting several points on the graph. By applying graphing functions to real-world situations, we can solve optimization problems, model real-world situations, and analyze data.

* Practice Graphing Functions: To practice graphing functions, try graphing different functions, such as $f(x)=x^2$ , $g(x)=(x-5)^2-9$ , and $h(x)=(x+2)^2+1$ .

* Explore Real-World Applications: To explore real-world applications of graphing functions, try modeling real-world situations, such as the motion of an object or the growth of a population.

* Analyze Data: To analyze data, try graphing different data sets and identifying patterns or trends.

References

Algebra and Calculus: Graphing functions is an essential skill in algebra and calculus.
Mathematics: Graphing functions has numerous applications in mathematics, science, and engineering.
Real-World Situations: Graphing functions can be used to model real-world situations, such as the motion of an object or the growth of a population.

Glossary

Parent Function: The basic function from which other functions are derived.
Horizontal Shift: A shift in the x-coordinate of a graph.
Vertical Shift: A shift in the y-coordinate of a graph.
Graphing Functions: The process of visualizing the relationship between the input and output values of a function.

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Introduction

Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. In this article, we will answer some frequently asked questions (FAQs) about graphing functions.

Q: What is a parent function?

A: A parent function is the basic function from which other functions are derived. For example, the parent function of $f(x)=x^2$ is $f(x)=x^2$ itself.

Q: What is a horizontal shift?

A: A horizontal shift is a shift in the x-coordinate of a graph. For example, to shift the graph of $f(x)=x^2$ right 5 units, we need to replace x with (x - 5) in the equation of $f(x)=x^2$ .

Q: What is a vertical shift?

A: A vertical shift is a shift in the y-coordinate of a graph. For example, to shift the graph of $f(x)=x^2$ down 9 units, we need to subtract 9 from the equation of $f(x)=x^2$ .

Q: How do I graph a function?

A: To graph a function, you need to plot several points on the graph and then connect them to form a smooth curve. You can use a graphing calculator or a computer program to graph a function.

Q: What are some common types of functions?

A: Some common types of functions include:

Linear functions: Functions of the form $f(x)=mx+b$ , where m and b are constants.
Quadratic functions: Functions of the form $f(x)=ax^2+bx+c$ , where a, b, and c are constants.
Polynomial functions: Functions of the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , where a_n, a_{n-1}, ..., a_1, and a_0 are constants.
Rational functions: Functions of the form $f(x)=\frac{p(x)}{q(x)}$ , where p(x) and q(x) are polynomials.

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

A: To determine the domain and range of a function, you need to consider the values of x and y that make the function undefined or undefined. For example, if a function has a denominator of zero, it is undefined at that point.

Q: What are some common graphing techniques?

A: Some common graphing techniques include:

Plotting points: Plotting several points on the graph and then connecting them to form a smooth curve.
Using a graphing calculator: Using a graphing calculator to graph a function.
Using a computer program: Using a computer program to graph a function.
Graphing transformations: Graphing transformations, such as horizontal and vertical shifts, to graph a function.

Q: How do I graph a function with a hole?

A: To graph a function with a hole, you need to plot the points on the graph that are not part of the hole and then connect them to form a smooth curve. You can use a graphing calculator or a computer program to graph a function with a hole.

Q: How do I graph a function with a vertical asymptote?

A: To graph a function with a vertical asymptote, you need to plot the points on the graph that are not part of the vertical asymptote and then connect them to form a smooth curve. You can use a graphing calculator or a computer program to graph a function with a vertical asymptote.

Q: How do I graph a function with a horizontal asymptote?

A: To graph a function with a horizontal asymptote, you need to plot the points on the graph that are not part of the horizontal asymptote and then connect them to form a smooth curve. You can use a graphing calculator or a computer program to graph a function with a horizontal asymptote.

Conclusion

In conclusion, graphing functions is an essential skill in mathematics, particularly in algebra and calculus. By understanding how to graph functions, you can visualize the relationship between the input and output values of a function. This can be achieved by plotting several points on the graph and then connecting them to form a smooth curve. By applying graphing functions to real-world situations, you can solve optimization problems, model real-world situations, and analyze data.

Final Thoughts

Graphing functions is a powerful tool that can be used to solve a wide range of problems in mathematics, science, and engineering. By understanding how to graph functions, you can visualize the relationship between the input and output values of a function and make informed decisions.

Introduction

Understanding the Parent Function

Characteristics of the Parent Function

* Vertex: (0, 0)

* Axis of Symmetry: x-axis

* Direction of Opening: Upwards

Shifting the Graph of the Parent Function

Horizontal Shift

* Right 5 units: To shift the graph of f(x)=x2f(x)=x^2f(x)=x2 right 5 units, we need to replace x with (x - 5) in the equation of f(x)=x2f(x)=x^2f(x)=x2. This gives us the equation f(x−5)=(x−5)2f(x-5)=(x-5)^2f(x−5)=(x−5)2.

Vertical Shift

* Down 9 units: To shift the graph of f(x)=x2f(x)=x^2f(x)=x2 down 9 units, we need to subtract 9 from the equation of f(x)=x2f(x)=x^2f(x)=x2. This gives us the equation g(x)=(x−5)2−9g(x)=(x-5)^2-9g(x)=(x−5)2−9.

Graphing the Function g(x)=(x−5)2−9g(x)=(x-5)^2-9g(x)=(x−5)2−9

Graphing the Function

* Plotting Points: To plot points on the graph, we need to substitute different values of x into the equation of g(x)=(x−5)2−9g(x)=(x-5)^2-9g(x)=(x−5)2−9 and calculate the corresponding values of y.

Example

Conclusion

Applications of Graphing Functions

* Optimization Problems: Graphing functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.

* Modeling Real-World Situations: Graphing functions can be used to model real-world situations, such as the motion of an object or the growth of a population.

* Analyzing Data: Graphing functions can be used to analyze data, such as the relationship between two variables.

Final Thoughts

* Practice Graphing Functions: To practice graphing functions, try graphing different functions, such as f(x)=x2f(x)=x^2f(x)=x2, g(x)=(x−5)2−9g(x)=(x-5)^2-9g(x)=(x−5)2−9, and h(x)=(x+2)2+1h(x)=(x+2)^2+1h(x)=(x+2)2+1.

* Explore Real-World Applications: To explore real-world applications of graphing functions, try modeling real-world situations, such as the motion of an object or the growth of a population.

* Analyze Data: To analyze data, try graphing different data sets and identifying patterns or trends.

References

Glossary

Introduction

Q: What is a parent function?

Q: What is a horizontal shift?

Q: What is a vertical shift?

Q: How do I graph a function?

Q: What are some common types of functions?

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

Q: What are some common graphing techniques?

Q: How do I graph a function with a hole?

Q: How do I graph a function with a vertical asymptote?

Q: How do I graph a function with a horizontal asymptote?

Conclusion

Final Thoughts

* Right 5 units: To shift the graph of $f(x)=x^2$ right 5 units, we need to replace x with (x - 5) in the equation of $f(x)=x^2$ . This gives us the equation $f(x-5)=(x-5)^2$ .

* Down 9 units: To shift the graph of $f(x)=x^2$ down 9 units, we need to subtract 9 from the equation of $f(x)=x^2$ . This gives us the equation $g(x)=(x-5)^2-9$ .

Graphing the Function $g(x)=(x-5)^2-9$

* Plotting Points: To plot points on the graph, we need to substitute different values of x into the equation of $g(x)=(x-5)^2-9$ and calculate the corresponding values of y.

* Practice Graphing Functions: To practice graphing functions, try graphing different functions, such as $f(x)=x^2$ , $g(x)=(x-5)^2-9$ , and $h(x)=(x+2)^2+1$ .