Use Technology To Find Points And Then Graph The Function Y = − ( X − 2 ) 2 + 5 Y=-(x-2)^2+5 Y = − ( X − 2 ) 2 + 5 , Following The Instructions Below.Plot At Least Five Points That Fit On The Axes Below. Click A Point To Delete It.

Introduction

In mathematics, 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 explore the graph of a specific quadratic function, $y = -(x-2)^2 + 5$, using technology to find points and then graph the function.

Understanding the Function

The given function is $y = -(x-2)^2 + 5$. This function can be rewritten as $y = -(x^2 - 4x + 4) + 5$, which simplifies to $y = -x^2 + 4x - 4 + 5$. Further simplifying, we get $y = -x^2 + 4x + 1$. This is a quadratic function in the form $y = ax^2 + bx + c$, where $a = -1$, $b = 4$, and $c = 1$.

Graphing the Function

To graph the function, we need to find at least five points that fit on the axes below. We can use technology, such as a graphing calculator or a computer algebra system, to find these points.

Finding Points

To find points on the graph, we can substitute different values of $x$ into the function and calculate the corresponding values of $y$. Let's find five points on the graph:

$x$	$y$
-2	9
-1	6
0	1
1	0
2	1

These points are plotted on the axes below.

Plotting Points

To plot the points, we can use a graphing calculator or a computer algebra system. We can also use a graphing software, such as Desmos or GeoGebra, to plot the points.

Analyzing the Graph

The graph of the function $y = -(x-2)^2 + 5$ is a parabola that opens downward. The vertex of the parabola is at the point $(2, 5)$. The parabola intersects the x-axis at the points $(-1, 0)$ and $(3, 0)$. The parabola also intersects the y-axis at the point $(0, 1)$.

Conclusion

In this article, we explored the graph of a quadratic function, $y = -(x-2)^2 + 5$, using technology to find points and then graph the function. We found five points on the graph and plotted them on the axes below. We also analyzed the graph and found the vertex, x-intercepts, and y-intercept of the parabola.

Discussion

What are some other ways to find points on a graph? How can we use technology to graph a function? What are some real-world applications of quadratic functions?

References

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

Additional Resources

• [1] "GeoGebra Graphing Software" by GeoGebra
• [2] "Mathway Algebra Solver" by Mathway
• [3] "Wolfram Alpha Calculator" by Wolfram Alpha
  Quadratic Function Graphing Q&A =====================================

Introduction

In our previous article, we explored the graph of a quadratic function, $y = -(x-2)^2 + 5$, using technology to find points and then graph the function. In this article, we will answer some frequently asked questions about quadratic function graphing.

Q&A

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 $y = ax^2 + bx + c$, 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 from decreasing to increasing or vice versa. It is also the minimum or maximum point of the graph.

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. Then, substitute this value of $x$ into the function to find the corresponding value of $y$.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points on the graph where the function intersects the x-axis. They are the solutions to the equation $y = 0$.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you can set the function equal to zero and solve for $x$. This will give you the x-coordinates of the x-intercepts.

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

A: The y-intercept of a quadratic function is the point on the graph where the function intersects the y-axis. It is the value of $y$ when $x = 0$.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you can substitute $x = 0$ into the function and solve for $y$.

Q: Can I graph a quadratic function by hand?

A: Yes, you can graph a quadratic function by hand using a table of values or by plotting points on a coordinate plane.

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

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

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Solving problems involving optimization
• Modeling population growth or decline

Conclusion

In this article, we answered some frequently asked questions about quadratic function graphing. We covered topics such as the vertex, x-intercepts, and y-intercept of a quadratic function, as well as real-world applications of quadratic functions.

Discussion

What are some other questions you have about quadratic function graphing? How can you apply quadratic functions to real-world problems?

References

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

Additional Resources

• [1] "GeoGebra Graphing Software" by GeoGebra
• [2] "Mathway Algebra Solver" by Mathway
• [3] "Wolfram Alpha Calculator" by Wolfram Alpha