1: Lee graphs the line y = 2x + 8 on a coordinate plane. What are the coordinates of the x-intercept of the line?
2: Data collected and plotted by Shawn has generated the quadratic equation y = -3x^2 - 6x + 4. What are the coordinates of the maximum of this function?
3: Hector graphs the line y = -2x + 3 on a coordinate plane. Jon graphs a line perpendicular to Hector's line, passing through the point (1,-2). What is the slope of Jon's line?
4: Juan plots two points on a coordinate plane: (-2,3) and (4,-1). What is the slope of the line that contains these two points?
any help would be greatly appreciated!!
Just to summarize what happened here.
qbqr21 plotted the graph for you and gave you the x-intercept, but how did he find it?
we find intercepts by the following.
note that the y-axis is a vertical line passing through x = 0, and the x-value of a point on the y-axis is always 0. similarly, the x-axis is a horizontal line passing through y = 0, and the y-value of any point on the x-axis is always 0. so,
to find the x-intercept we set y = 0
to find the y-intercept we set x = 0
now we were given y = 2x + 8 .......the coefficient of x is the slope and the lone constant is the y-intercept, why?
well, for y-intercept, set x = 0, we get
y = 2(0) + 8 = 8
so the y-intercept is (0,8)
for x-intercept, set y = 0
=> 0 = 2x + 8
=> 2x = -8
=> x = -4
so the x-intercept is (-4,0), which is what qbkr21 got.
this is a downward opening parabola (do you see why?), it has a maximum at it's vertex--the highest point.2: Data collected and plotted by Shawn has generated the quadratic equation y = -3x^2 - 6x + 4. What are the coordinates of the maximum of this function?
to find the x-value for the vertex of a parabola, whether it opens up or down, we solve the equation:
x = -b/2a
where b is the coefficient of x and a is the coefficient of x^2
note: if we had a positive coefficient for the x^2 we would have an upward opening parabola, and the vertex would give it's minimum point
in this problem, we are given: y = -3x^2 - 6x + 4, the coefficient of x^2 is -3 which we call a, and the coefficient of x is -6 which we call b. so for the vertex:
x = -b/2a = 6/2(-3) = -1
when x is -1,
y = -3(-1)^2 - 6(-1) + 4 = 7
so the maximum point is (-1, 7), which is what qbkr21 got
Challenge: try to find the x and y-intercepts for this function.
now we study the relation of the slopes of two lines.3: Hector graphs the line y = -2x + 3 on a coordinate plane. Jon graphs a line perpendicular to Hector's line, passing through the point (1,-2). What is the slope of Jon's line?
to reuse the variables topsquark used, let the slopes of two lines be m1 and m2
two lines are parallel if they have the same slope, that is m1 = m2
two lines are perpendicular if their slopes are the negative inverses of each other, that is m1 = -1/m2. basically, if we take one slope, turn it upside down and attach a minus sign in fron of it, we will get the other slope
topsquark did the calculations for this, so i won't repeat it.
the conventional variable to represent the slope of a line is m. we call a function a straight line if it can be written in the form:4: Juan plots two points on a coordinate plane: (-2,3) and (4,-1). What is the slope of the line that contains these two points?
y = mx + b, where m is the slope and b is the y-intercept.
the slope is defined as the ratio of the rise over run, that is, it is a measure of the rate at which the graph is increasing or decreasing. we find it by measuring how high we go between one point and another divided by the corresponding distance we travel. the formula for the slope is as follows:
let two coordinate points be (x1,y1) and (x2,y2)
the slope of the line connecting these points is given by:
m = (y2 - y1)/(x2 - x1) ............(change in height)/(change in horizontal distance travelled)
topsquark did this for you, so again, i won't repeat it
TPH added the comment that this is true only if x1 not= x2, since that would cause or slope to be undefined, as it would result in dividing by zero, which we can't do
Hope all that rambling helped