2.1.Sketch The Graph Of $g(x)=x^2-4x+4$. Plot The Values From The Table On The Set Of Axes Provided.\[\begin{tabular}{|c|c|c|c|c|c|}\hline & -1 & 0 & 1 & 2 & 4 \\\hline$g(x$\] & 9 & 4 & 1 & 0 & 4 \\\hline\end{tabular}\]Use The Graph To

Introduction

In mathematics, graphing quadratic functions is an essential skill that helps us visualize and understand the behavior of these functions. In this article, we will focus on sketching the graph of the quadratic function $g(x) = x^2 - 4x + 4$. We will use a table of values to plot the graph on a set of axes and then use the graph to analyze the function's behavior.

Understanding the Quadratic Function

A quadratic function is a polynomial function of degree two, which means it has the general form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

In our case, the quadratic function is $g(x) = x^2 - 4x + 4$. To understand the behavior of this function, we need to analyze its components. The coefficient of the $x^2$ term is 1, which means the parabola opens upwards. The coefficient of the $x$ term is -4, which means the parabola is shifted to the right. The constant term is 4, which means the parabola is shifted upwards.

Plotting the Values from the Table

To plot the graph of the quadratic function, we need to use a table of values. The table provided shows the values of $x$ and the corresponding values of $g(x)$.

$x$	$g(x)$
-1	9
0	4
1	1
2	0
4	4

To plot the graph, we need to plot the points $(x, g(x))$ on the set of axes. We can start by plotting the point $(0, 4)$, which is the vertex of the parabola. Then, we can plot the points $(-1, 9)$, $(1, 1)$, $(2, 0)$, and $(4, 4)$.

Sketching the Graph

Once we have plotted the points, we can sketch the graph of the quadratic function. The graph is a parabola that opens upwards, with its vertex at the point $(0, 4)$. The parabola is shifted to the right due to the negative coefficient of the $x$ term, and it is shifted upwards due to the positive constant term.

Analyzing the Graph

The graph of the quadratic function provides valuable information about the function's behavior. We can see that the parabola has a minimum point at the vertex, which is the point $(0, 4)$. The parabola is symmetric about the vertical line $x = 0$, which means that the function is even.

We can also see that the parabola intersects the $x$-axis at the point $(2, 0)$. This means that the function has a root at $x = 2$, which is the value of $x$ that makes the function equal to zero.

Conclusion

In this article, we have sketched the graph of the quadratic function $g(x) = x^2 - 4x + 4$ using a table of values. We have analyzed the graph to understand the behavior of the function, including its vertex, symmetry, and roots. The graph provides valuable information about the function's behavior and can be used to make predictions about the function's values.

Key Takeaways

• The graph of a quadratic function is a parabola that opens upwards or downwards.
• The vertex of a parabola is the point where the function has a minimum or maximum value.
• The parabola is symmetric about the vertical line $x = 0$ if the function is even.
• The parabola intersects the $x$-axis at the points where the function has roots.

Further Reading

For further reading on quadratic functions and graphing, we recommend the following resources:

• Wikipedia: Quadratic function
• Math Open Reference: Quadratic function
• Khan Academy: Quadratic functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline
  Quadratic Function Graphing: Frequently Asked Questions ===========================================================

Introduction

Graphing quadratic functions is an essential skill in mathematics, and it can be a bit challenging for some students. In this article, we will answer some frequently asked questions about graphing quadratic functions, including questions about the shape of the graph, the location of the vertex, and the behavior of the function.

Q: What is the shape of the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the function has a minimum or maximum value. It is the lowest or highest point on the graph.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula:

x = -b / 2a

where a and b are the coefficients of the quadratic function.

Q: What is the x-intercept of a parabola?

A: The x-intercept of a parabola is the point where the graph intersects the x-axis. It is the value of x that makes the function equal to zero.

Q: How do I find the x-intercept of a parabola?

A: To find the x-intercept of a parabola, you can set the function equal to zero and solve for x.

Q: What is the y-intercept of a parabola?

A: The y-intercept of a parabola is the point where the graph intersects the y-axis. It is the value of y that makes the function equal to zero when x is equal to zero.

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

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

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

A: Yes, a parabola can have more than one x-intercept. This occurs when the function has multiple roots.

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

A: No, a parabola cannot have more than one y-intercept. The y-intercept is a single point on the graph.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a table of values to plot the points on the graph. You can also use the vertex form of the function to graph the parabola.

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: How do I use the vertex form to graph a parabola?

A: To use the vertex form to graph a parabola, you can substitute the values of a, h, and k into the formula and plot the points on the graph.

Conclusion

Graphing quadratic functions can be a bit challenging, but with practice and patience, you can become proficient in graphing these functions. Remember to use a table of values to plot the points on the graph, and use the vertex form to graph the parabola. With these tools, you can graph any quadratic function.

Key Takeaways

• The graph of a quadratic function is a parabola that opens upwards or downwards.
• The vertex of a parabola is the point where the function has a minimum or maximum value.
• The x-intercept of a parabola is the point where the graph intersects the x-axis.
• The y-intercept of a parabola is the point where the graph intersects the y-axis.
• A parabola can have more than one x-intercept, but it cannot have more than one y-intercept.

Further Reading

For further reading on quadratic functions and graphing, we recommend the following resources:

• Wikipedia: Quadratic function
• Math Open Reference: Quadratic function
• Khan Academy: Quadratic functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline