A line intersects the graph of y= x^2 twice. One point has an x- coordinate of -4, and the other point has an x-coordinate of 2.
A) Draw a sketch of both graphs, and find the equation of the line.
B) Find the measure of the angle that the line makes with the x-axis.
done on Monday
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A) Well, I would love to show you a sketch, but since I'm just a text-based bot, I don't have any artistic skills. However, I can help you find the equation of the line!
We have two points on the line: (-4, y) and (2, y). To find the equation, we'll use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
First, we need to find the slope, which is the change in y divided by the change in x. Since the line passes through (-4, y) and (2, y), the change in y is 0 for this line. And the change in x is 2 - (-4) = 6.
So, the slope (m) is 0 divided by 6, which is 0. Since the line is horizontal, it has a slope of 0.
Now, we can plug in one of the points, let's use (-4, y), into the slope-intercept form to find the value of b. We have y = 0 * (-4) + b, which simplifies to y = b.
Therefore, the equation of the line is y = b, where b can be any constant.
B) Since the line is horizontal and parallel to the x-axis, it doesn't make any angle with the x-axis. So, the measure of the angle that the line makes with the x-axis is 0 degrees. It's as straight as can be!
I hope my answers brought a smile to your face!
A) To draw a sketch of the graphs, we first plot the points given: (-4, 16) and (2, 4).
Next, we plot the graph of y = x^2, which is a parabola opening upwards. Since the y-value is equal to x^2, we can find the y-coordinate of each point by squaring the x-coordinate.
Now, we draw a line passing through these two points.
To find the equation of the line, we can use the point-slope form of a line. The formula is:
y - y1 = m(x - x1)
Let's choose the first point (-4, 16) as (x1, y1) and the second point (2, 4) as (x2, y2).
Using the slope formula, m = (y2 - y1) / (x2 - x1), we can calculate the slope:
m = (4 - 16) / (2 - (-4)) = -12 / 6 = -2
Now, substituting the values into the point-slope form:
y - 16 = -2(x - (-4))
y - 16 = -2(x + 4)
y - 16 = -2x - 8
y = -2x + 8
Therefore, the equation of the line passing through the points of intersection is y = -2x + 8.
B) To find the angle that the line makes with the x-axis, we determine the slope of the line and take the arctan of the slope.
The slope of the line is -2, which means it moves down 2 units for every 1 unit it moves to the right.
Using the arctan function, we can find the angle:
θ = arctan(-2)
Using a calculator, we find that the angle is approximately -63.43 degrees or around 243.43 degrees measured clockwise from the positive x-axis.
Therefore, the measure of the angle that the line makes with the x-axis is approximately 63.43 degrees or around 296.57 degrees measured counterclockwise from the positive x-axis.
To solve this problem, let's break it down into two parts:
A) Drawing the graphs and finding the equation of the line:
1. Start by plotting the given points: (-4, y) and (2, y) on a coordinate plane.
2. Draw the graph of the quadratic function y = x^2. It will be a U-shaped curve symmetric about the y-axis.
3. Since the line intersects the graph of y = x^2 at two points, draw a straight line passing through the given points (-4, y) and (2, y). This line will intersect the curve at those two points.
4. Now we need to find the equation of this line. To do that, we can use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
- Choose either of the given points, let's say (2, y), so x1 = 2 and y1 = y.
- To find the slope (m), we can use the formula: m = (y2 - y1) / (x2 - x1), where (x2, y2) is the other given point (-4, y).
- Substituting the values: m = (y - y) / (-4 - 2) = 0 / -6 = 0.
- Since the slope is 0, the equation of the line is y = y1, where y1 is the y-coordinate of the chosen point.
- In this case, y = y1 = y (since both points have the same y-coordinate value).
B) Finding the angle that the line makes with the x-axis:
1. From the equation y = y1, we can see that the line is parallel to the x-axis.
2. The angle that a line makes with the x-axis when it is parallel to it is 0 degrees.
3. Therefore, the measure of the angle that the line makes with the x-axis is 0 degrees.
I hope this helps! Let me know if you have any further questions.