Finding a and b for y = ax^2 + bx Passing Through (2,4) with Gradient 8

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SUMMARY

The problem involves finding the coefficients a and b in the quadratic equation y = ax² + bx, given that the curve passes through the point (2, 4) and has a gradient of 8 at that point. The first equation derived from the point is 4a + 2b = 4, while the second equation from the gradient is 4a + b = 8. Solving these simultaneous equations yields a = 3 and b = -4, resulting in the equation y = 3x² - 4x.

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Homework Statement



The curve y = ax^{2} + bx passes through the point (2,4) with gradient 8. Find a and b .

I have no idea how to work out a and b , do I use simultaneous equations?

Homework Equations


The Attempt at a Solution



I have no idea how to work out a and b , do I use simultaneous equations?
 
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well start of by thinking about what your dy/dx would be...
 
okay so dy/dx = 2ax + b from this how do I find what a and b equal ?
 
Use the fact that (2, 4) is a point on the curve in your given equation, which will give you an equation that involves only a and b.

Then use the fact that at x = 2, dy/dx = 8 in your equation for the derivative. This will give you another equation in a and b.

Finally, solve the two equations in a and b simultaneously.
 
Mark44 said:
Use the fact that (2, 4) is a point on the curve in your given equation, which will give you an equation that involves only a and b.

Then use the fact that at x = 2, dy/dx = 8 in your equation for the derivative. This will give you another equation in a and b.

Finally, solve the two equations in a and b simultaneously.

so I just sub in the values for x and y ?

are these two equations correct..

4a + b = 4

4a + b = 8

?
 
Have another look at the first equation.
 
danago said:
Have another look at the first equation.

x=2 so 2x2=4? can you give more hints, cause I am really stuck

thank you!
 
is it meant to be 32a + b =4 ?
 
tweety1234 said:
is it meant to be 32a + b =4 ?
No.

Write your first equation.
Substitute 2 for x and 4 for y in that equation. The other equation you wrote, 4a + b = 8, is correct.
 
  • #10
y = ax^{2} + bx , 4a + 2b = 4 is this correct? so a = 3 and b = -4 ?
 
  • #11
Check it yourself. IF a= 3 and b= -4, you have y= 3x2- 4x. What is y when x= 2? What is y' when x= 2?
 

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