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. Find the equation of the line.
B. Find the measure of the angle that the line makes with the x-axis.
when x=-4, y = 16
when x = 2, y = 4
two intersection points are (-4,16) and (2,4)
slope = (4-16)/(2+4) = -12/6 = -2
equation of line:
y = -2x + b, but (2,4) lies on it, so
4 = -4 + b
b = 8
equation of line:
y = -2x + 8
a line with a slope of -2 makes and angle of
tan^-1 (-2) or 116.565°
check: tan 116.565 = -2
Thank you very very much, I really appreciate this.
A. To find the equation of the line, we can use the two given points that it intersects the graph of y=x^2. Let's denote the first point as A(-4, y1) and the second point as B(2, y2).
We can find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values:
m = (y2 - y1) / (2 - (-4))
m = (y2 - y1) / 6
Since the line intersects the graph of y=x^2 twice, it means it is a tangent line to the curve. A tangent line to the curve at a given point has the same slope as the derivative of the curve at that point.
The derivative of y=x^2 is y' = 2x. So at the point (-4, y1), the derivative of y=x^2 will have the same slope as the line. Therefore:
2x = (y2 - y1) / 6
Solve for y2 - y1:
y2 - y1 = 12x
Now, we can substitute the values of x and y for one of the given points (let's use point A: -4, y1):
y2 - y1 = 12(-4)
y2 - y1 = -48
Therefore, the equation of the line is:
y2 = y1 - 48
B. To find the measure of the angle that the line makes with the x-axis, we need to find the slope of the line. We already found the slope in part A, which is:
m = (y2 - y1) / 6
Since the line is intersecting the graph of y=x^2 twice, it is tangent to the curve at those points. The slope of the tangent line at a given point is equal to the derivative of the curve at that point.
The derivative of y=x^2 is y' = 2x. So the slope of the line is:
m = 2x / 6
m = x / 3
The angle that a line makes with the x-axis is given by the arctan of the slope of the line.
Angle = arctan(m)
Angle = arctan(x / 3)
To find the measure of the angle, substitute the value of x for one of the given points (let's use point A: -4, y1):
Angle = arctan(-4 / 3)
Using a calculator, we find that the measure of the angle is approximately -53.13 degrees.
To find the equation of the line, we can use the two points given: (-4, ) and (2, ).
A. Finding the slope (m):
The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the formula: m = (y2 - y1) / (x2 - x1).
Using the given points, we have:
m = ( - ) / ( - ).
B. Finding the y-intercept (b):
To find the y-intercept, we can use the slope-intercept form of a line, which is y = mx + b. Since the line passes through the point (2, ), we can substitute the coordinates into the equation to solve for b.
Using the point (2, ), we have:
= (2) + b.
Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line in slope-intercept form.
Using the obtained values, we get the equation: y = x +
For Part B, finding the measure of the angle that the line makes with the x-axis, we need to determine the slope of the line.
In the equation of the line y = x + , the coefficient of the x-term is the slope. In this case, the slope is 1.
The angle that a line makes with the x-axis is given by the inverse tangent (arctan) of the slope. Therefore, the measure of the angle can be found by taking the inverse tangent of 1.
Using a calculator or a math software, calculate arctan(1) to find the angle measure in radians or use the arctan function in degrees mode to find the angle measure in degrees.
Therefore, the measure of the angle that the line makes with the x-axis is radians (or approximately degrees).