# Thread: help with a few simple problems..

1. ## help with a few simple problems..

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!!

2. ## Re:

Originally Posted by outlined
1: Lee graphs the line y = 2x + 8 on a coordinate plane. What are the coordinates of the x-intercept of the line?

Thus

#1. (-4,0)

Graph y=2x+8

3. ## Re:

Originally Posted by outlined

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?
Graph

y = -3x^2 - 6x + 4

Thus we see our Maximum is at (-1,7)

4. Originally Posted by outlined
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?
The slopes of perpendicular lines are related by
m2 = -1/m1

So the slope of Jon's line is
m = -1/(-2) = 1/2

-Dan

5. Originally Posted by outlined
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?
Given two points (x1, y1) and (x2, y2) the slope of the line that passes through them is defined to be:
m = (y2 - y1)/(x2 - x1)
(The points may be taken to be in any order.)

So
m = (-1 - 3)/(4 - (-2)) = (-4)/(6) = -2/3

-Dan

6. Originally Posted by topsquark
(The points may be taken to be in any order.)
With x1!=x2

7. Originally Posted by ThePerfectHacker
With x1!=x2
But of course!

-Dan

8. thank you all so much!! i'm horrible at these things. =)

9. Just to summarize what happened here.

Originally Posted by outlined
1: Lee graphs the line y = 2x + 8 on a coordinate plane. What are the coordinates of the x-intercept of the line?
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.

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?
this is a downward opening parabola (do you see why?), it has a maximum at it's vertex--the highest point.

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.

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?
now we study the relation of the slopes of two lines.

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.

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?
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:

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