Finding the equation of a line

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In summary, the conversation is about a question involving finding the values of constants in a curve equation given certain points and information about the gradient. Kafka has made several attempts but is unsure how to approach one part of the equation. Through their attempts, they have determined the values of c and b and used a point to find the value of a.
  • #1
Kafka
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Hi there,

I have to complete the following question, but I have no idea how to approach it (there are four other parts to the question that I can't complete until I get the first bit). I have made numerous attempts, but am not sure how to get the a part of the equation.

All help would be appreciated!

Thanks,
Kafka

Question
A curve f(x) is defined by the equation : y = ax² + bx + c, where a, b and c are constants.

The curve crosses the y-axis at the point (0,4). At this point the gradient of the graph is -5.

The curve crosses the x-axis at point (1,0).

(i) Find the values of a, b, and c and write down the equation of the curve

Attempt

Sub point (0.4) into equation to get c (x=0,y=4):

y = ax² + bx + c
4 = 0 + 0 + c
c = 4

If gradient at point (0,4) is -5, then dy/dx must be equal to -5.

dy/dx = 2ax + b
-5 = 2ax + b
-5 = 2a(0) + b
-5 = 0 + b
b = -5

*not sure about the bit below*

c=-5, b=4, so sub these into equation of curve and use a point to find a

y = ax² + bx + c

at (1,0) x=1, y=0

y = ax² + bx + c
0 = a1² + (-5 x 1) + 4
0 = a -5 + 4
0 = a - 1
a = 1 ?
 
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  • #2
Looks good to me :approve:
 
  • #3
Now if only PHYSICS students will recognize this as a projectile problem...
 
  • #4
Kafka, you did that exactly right.

It does, however, have nothing to do with "differential equations" so I am moving it to the Calculus section.
 

What is the equation of a line?

The equation of a line is a mathematical expression that represents a straight line on a graph. It is typically written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

How do you find the equation of a line?

To find the equation of a line, you need to know two important pieces of information: the slope of the line and the y-intercept. Once you have these values, you can plug them into the equation y = mx + b to get the equation of the line.

What is the slope of a line?

The slope of a line is a measure of how steep the line is. It is calculated by dividing the change in the y-coordinates by the change in the x-coordinates between any two points on the line. The slope can also be represented by the letter "m" in the equation y = mx + b.

How do you find the slope of a line?

To find the slope of a line, you can use the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line. Alternatively, you can also use the rise over run method, where you count the change in y-coordinates (rise) and the change in x-coordinates (run) between two points on the line.

What is the y-intercept of a line?

The y-intercept of a line is the point where the line crosses the y-axis on a graph. It is represented by the letter "b" in the equation y = mx + b. The y-intercept is important because it gives information about the starting point of the line and its relationship with the y-axis.

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