(b) On The Grid, Draw The Graph Of Y = 8 − 2 X Y = 8 - 2x Y = 8 − 2 X For 0 ⩽ X ⩽ 4 0 \leqslant X \leqslant 4 0 ⩽ X ⩽ 4 .Find The Coordinates Of The Point Where The Line Intersects The Graph Of Y = 8 Y = 8 Y = 8 .Complete The Table Of Values For $y = X^2 - 4x -

Introduction

In this article, we will delve into the world of graphing and intersections, exploring the concepts of linear and quadratic functions. We will begin by graphing the line $y = 8 - 2x$ for the interval $0 \leqslant x \leqslant 4$. Then, we will find the coordinates of the point where this line intersects the graph of $y = 8$. Finally, we will complete the table of values for the quadratic function $y = x^2 - 4x$.

Graphing the Line $y = 8 - 2x$

To graph the line $y = 8 - 2x$, we can start by finding the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is $(0, 8)$, since when $x = 0$, $y = 8$. Next, we can find the x-intercept, which is the point where the line intersects the x-axis. To do this, we set $y = 0$ and solve for $x$. This gives us:

$$0 = 8 - 2x$$

Solving for $x$, we get:

$$2x = 8$$

$$x = 4$$

So, the x-intercept is $(4, 0)$.

Now that we have the y-intercept and x-intercept, we can use these points to graph the line. We can also use the slope of the line, which is $-2$, to help us graph it. The slope tells us how steep the line is, and it can be used to draw the line between the two intercepts.

Finding the Intersection with $y = 8$

To find the coordinates of the point where the line intersects the graph of $y = 8$, we can set the two equations equal to each other and solve for $x$. This gives us:

$$8 - 2x = 8$$

Solving for $x$, we get:

$$-2x = 0$$

$$x = 0$$

So, the line intersects the graph of $y = 8$ at the point $(0, 8)$.

Completing the Table of Values for $y = x^2 - 4x$

To complete the table of values for the quadratic function $y = x^2 - 4x$, we can start by finding the values of $y$ for different values of $x$. We can use the equation $y = x^2 - 4x$ to find the values of $y$.

$x$	$y = x^2 - 4x$
0	0
1	-3
2	0
3	3
4	0

We can see that the table of values shows a pattern, with the values of $y$ increasing and then decreasing as $x$ increases. This is because the quadratic function $y = x^2 - 4x$ has a parabolic shape, with a vertex at the point $(2, 0)$.

Conclusion

In this article, we have explored the concepts of graphing and intersections, using the line $y = 8 - 2x$ and the quadratic function $y = x^2 - 4x$ as examples. We have graphed the line and found the coordinates of the point where it intersects the graph of $y = 8$. We have also completed the table of values for the quadratic function, showing the pattern of the values of $y$ as $x$ increases.

Graphing the Line $y = 8 - 2x$

To graph the line $y = 8 - 2x$, we can start by finding the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is $(0, 8)$, since when $x = 0$, $y = 8$. Next, we can find the x-intercept, which is the point where the line intersects the x-axis. To do this, we set $y = 0$ and solve for $x$. This gives us:

$$0 = 8 - 2x$$

Solving for $x$, we get:

$$2x = 8$$

$$x = 4$$

So, the x-intercept is $(4, 0)$.

Now that we have the y-intercept and x-intercept, we can use these points to graph the line. We can also use the slope of the line, which is $-2$, to help us graph it. The slope tells us how steep the line is, and it can be used to draw the line between the two intercepts.

Finding the Intersection with $y = 8$

To find the coordinates of the point where the line intersects the graph of $y = 8$, we can set the two equations equal to each other and solve for $x$. This gives us:

$$8 - 2x = 8$$

Solving for $x$, we get:

$$-2x = 0$$

$$x = 0$$

So, the line intersects the graph of $y = 8$ at the point $(0, 8)$.

Completing the Table of Values for $y = x^2 - 4x$

To complete the table of values for the quadratic function $y = x^2 - 4x$, we can start by finding the values of $y$ for different values of $x$. We can use the equation $y = x^2 - 4x$ to find the values of $y$.

$x$	$y = x^2 - 4x$
0	0
1	-3
2	0
3	3
4	0

We can see that the table of values shows a pattern, with the values of $y$ increasing and then decreasing as $x$ increases. This is because the quadratic function $y = x^2 - 4x$ has a parabolic shape, with a vertex at the point $(2, 0)$.

Conclusion

In this article, we have explored the concepts of graphing and intersections, using the line $y = 8 - 2x$ and the quadratic function $y = x^2 - 4x$ as examples. We have graphed the line and found the coordinates of the point where it intersects the graph of $y = 8$. We have also completed the table of values for the quadratic function, showing the pattern of the values of $y$ as $x$ increases.

Graphing and Intersections: A Mathematical Exploration

Graphing the Line $y = 8 - 2x$

To graph the line $y = 8 - 2x$, we can start by finding the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is $(0, 8)$, since when $x = 0$, $y = 8$. Next, we can find the x-intercept, which is the point where the line intersects the x-axis. To do this, we set $y = 0$ and solve for $x$. This gives us:

$$0 = 8 - 2x$$

Solving for $x$, we get:

$$2x = 8$$

$$x = 4$$

So, the x-intercept is $(4, 0)$.

Now that we have the y-intercept and x-intercept, we can use these points to graph the line. We can also use the slope of the line, which is $-2$, to help us graph it. The slope tells us how steep the line is, and it can be used to draw the line between the two intercepts.

Finding the Intersection with $y = 8$

To find the coordinates of the point where the line intersects the graph of $y = 8$, we can set the two equations equal to each other and solve for $x$. This gives us:

$$8 - 2x = 8$$

Solving for $x$, we get:

$$-2x = 0$$

$$x = 0$$

So, the line intersects the graph of $y = 8$ at the point $(0, 8)$.

Completing the Table of Values for $y = x^2 - 4x$

To complete the table of values for the quadratic function $y = x^2 - 4x$, we can start by finding the values of $y$ for different values of $x$. We can use the equation $y = x^2 - 4x$ to find the values of $y$.

$x$	$y = x^2 - 4x$
0	0
1	-3
2	0
3	3
4	0

$$

Q&A: Graphing and Intersections

Q: What is the y-intercept of the line $y = 8 - 2x$?

A: The y-intercept of the line $y = 8 - 2x$ is $(0, 8)$, since when $x = 0$, $y = 8$.

Q: What is the x-intercept of the line $y = 8 - 2x$?

A: The x-intercept of the line $y = 8 - 2x$ is $(4, 0)$, since when $y = 0$, $x = 4$.

Q: How do you find the intersection of the line $y = 8 - 2x$ and the graph of $y = 8$?

A: To find the intersection of the line $y = 8 - 2x$ and the graph of $y = 8$, we can set the two equations equal to each other and solve for $x$. This gives us:

$$8 - 2x = 8$$

Solving for $x$, we get:

$$-2x = 0$$

$$x = 0$$

So, the line intersects the graph of $y = 8$ at the point $(0, 8)$.

Q: How do you complete the table of values for the quadratic function $y = x^2 - 4x$?

A: To complete the table of values for the quadratic function $y = x^2 - 4x$, we can start by finding the values of $y$ for different values of $x$. We can use the equation $y = x^2 - 4x$ to find the values of $y$.

$x$	$y = x^2 - 4x$
0	0
1	-3
2	0
3	3
4	0

We can see that the table of values shows a pattern, with the values of $y$ increasing and then decreasing as $x$ increases.

Q: What is the vertex of the parabola $y = x^2 - 4x$?

A: The vertex of the parabola $y = x^2 - 4x$ is $(2, 0)$, since the parabola opens upward and the vertex is the lowest point on the parabola.

Q: How do you graph the line $y = 8 - 2x$?

A: To graph the line $y = 8 - 2x$, we can start by finding the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is $(0, 8)$, since when $x = 0$, $y = 8$. Next, we can find the x-intercept, which is the point where the line intersects the x-axis. To do this, we set $y = 0$ and solve for $x$. This gives us:

$$0 = 8 - 2x$$

Solving for $x$, we get:

$$2x = 8$$

$$x = 4$$

So, the x-intercept is $(4, 0)$.

Now that we have the y-intercept and x-intercept, we can use these points to graph the line. We can also use the slope of the line, which is $-2$, to help us graph it. The slope tells us how steep the line is, and it can be used to draw the line between the two intercepts.

Q: What is the slope of the line $y = 8 - 2x$?

A: The slope of the line $y = 8 - 2x$ is $-2$, since the line is decreasing at a rate of 2 units for every 1 unit increase in $x$.

Q: How do you find the equation of the line that passes through the points $(0, 8)$ and $(4, 0)$?

A: To find the equation of the line that passes through the points $(0, 8)$ and $(4, 0)$, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We can find the slope by using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points. Plugging in the values, we get:

$$m = \frac{0 - 8}{4 - 0}$$

$$m = -2$$

Now that we have the slope, we can find the y-intercept by plugging in one of the points into the equation. Let's use the point $(0, 8)$:

$$8 = -2(0) + b$$

$$b = 8$$

So, the equation of the line is $y = -2x + 8$.

Q: What is the equation of the line that passes through the points $(0, 8)$ and $(4, 0)$?

A: The equation of the line that passes through the points $(0, 8)$ and $(4, 0)$ is $y = -2x + 8$.