Use Technology To Find Points And Then Graph The Function $y=(x-2)^2-1$, Following The Instructions Below.Plot At Least Five Points That Fit On The Axes Below. Click A Point To Delete It.

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Introduction

Graphing a quadratic function can be a complex task, especially when dealing with equations that involve variables and constants. However, with the help of technology, we can easily plot points and graph the function. In this article, we will explore how to use technology to find points and then graph the function $y=(x-2)^2-1$.

Understanding the Function

Before we begin, let's take a closer look at the function $y=(x-2)^2-1$. This is a quadratic function in the form of $y=a(x-h)^2+k$, where $a$ is the coefficient of the squared term, $h$ is the horizontal shift, and $k$ is the vertical shift.

In this case, the function has a coefficient of $1$, a horizontal shift of $2$, and a vertical shift of $-1$. This means that the graph of the function will be a parabola that opens upwards, shifted $2$ units to the right and $1$ unit down.

Plotting Points

To plot points on the graph, we need to find the coordinates of the points that fit on the axes. We can do this by plugging in values of $x$ into the function and solving for $y$. Let's start by finding five points that fit on the axes.

Point 1

Let's start by finding the point when $x=0$. Plugging in $x=0$ into the function, we get:

y=(0−2)2−1y=(0-2)^2-1

y=4−1y=4-1

y=3y=3

So, the point when $x=0$ is $(0,3)$.

Point 2

Next, let's find the point when $x=1$. Plugging in $x=1$ into the function, we get:

y=(1−2)2−1y=(1-2)^2-1

y=1−1y=1-1

y=0y=0

So, the point when $x=1$ is $(1,0)$.

Point 3

Now, let's find the point when $x=3$. Plugging in $x=3$ into the function, we get:

y=(3−2)2−1y=(3-2)^2-1

y=1−1y=1-1

y=0y=0

So, the point when $x=3$ is $(3,0)$.

Point 4

Next, let's find the point when $x=4$. Plugging in $x=4$ into the function, we get:

y=(4−2)2−1y=(4-2)^2-1

y=4−1y=4-1

y=3y=3

So, the point when $x=4$ is $(4,3)$.

Point 5

Finally, let's find the point when $x=5$. Plugging in $x=5$ into the function, we get:

y=(5−2)2−1y=(5-2)^2-1

y=9−1y=9-1

y=8y=8

So, the point when $x=5$ is $(5,8)$.

Graphing the Function

Now that we have found five points that fit on the axes, we can use technology to graph the function. We can use a graphing calculator or a computer program to plot the points and draw the graph.

Here is an example of what the graph might look like:

Graph of the function

As we can see, the graph of the function is a parabola that opens upwards, shifted $2$ units to the right and $1$ unit down. The points we found earlier are marked on the graph, and we can see that they fit perfectly on the axes.

Conclusion

In this article, we used technology to find points and then graph the function $y=(x-2)^2-1$. We started by understanding the function and its properties, and then we found five points that fit on the axes. Finally, we used technology to graph the function and see the results.

Graphing a quadratic function can be a complex task, but with the help of technology, we can easily plot points and graph the function. This is especially useful for students who are learning about quadratic functions and need to visualize the graph.

Future Work

In the future, we can use technology to explore other properties of the function, such as its vertex and axis of symmetry. We can also use technology to graph other types of functions, such as linear and polynomial functions.

References

  • [1] "Graphing Quadratic Functions" by Math Open Reference
  • [2] "Quadratic Functions" by Khan Academy
  • [3] "Graphing Functions" by Wolfram Alpha

Appendix

Here are the coordinates of the five points we found earlier:

Point x y
1 0 3
2 1 0
3 3 0
4 4 3
5 5 8

Introduction

Graphing quadratic functions can be a complex task, but with the help of technology, we can easily plot points and graph the function. In this article, we will answer some frequently asked questions about graphing quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that 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.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes direction. It is the minimum or maximum point of the graph, depending on whether the function opens upwards or downwards.

Q: How do 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. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can plug the x-coordinate back into the function.

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. It is the line of symmetry of the graph, and it divides the graph into two equal parts.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program to plot points on the graph. You can also use the vertex and axis of symmetry to help you graph the function.

Q: What are some common mistakes to avoid when graphing quadratic functions?

A: Some common mistakes to avoid when graphing quadratic functions include:

  • Not using a graphing calculator or computer program to plot points on the graph
  • Not finding the vertex and axis of symmetry of the function
  • Not using the correct formula to find the vertex
  • Not checking the graph for accuracy

Q: How can I use technology to graph quadratic functions?

A: There are many ways to use technology to graph quadratic functions, including:

  • Using a graphing calculator to plot points on the graph
  • Using a computer program to graph the function
  • Using a spreadsheet to graph the function
  • Using a graphing app on a smartphone or tablet

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

A: Quadratic functions have many real-world applications, including:

  • Modeling the motion of objects under the influence of gravity
  • Finding the maximum or minimum value of a function
  • Solving problems involving optimization
  • Modeling population growth or decline

Q: How can I use quadratic functions to solve problems?

A: Quadratic functions can be used to solve a wide range of problems, including:

  • Finding the maximum or minimum value of a function
  • Solving problems involving optimization
  • Modeling population growth or decline
  • Finding the area or perimeter of a shape

Conclusion

Graphing quadratic functions can be a complex task, but with the help of technology, we can easily plot points and graph the function. By understanding the properties of quadratic functions and using technology to graph them, we can solve a wide range of problems and model real-world situations.

References

  • [1] "Graphing Quadratic Functions" by Math Open Reference
  • [2] "Quadratic Functions" by Khan Academy
  • [3] "Graphing Functions" by Wolfram Alpha

Appendix

Here are some additional resources for learning about quadratic functions:

  • [1] "Quadratic Functions" by Math Is Fun
  • [2] "Graphing Quadratic Functions" by Purplemath
  • [3] "Quadratic Functions" by IXL