Equation of a curve passing through (1,2)

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In summary, the curve with slope 2x + 3 at each point (x, y) and passing through the point (1,2) can be represented by the equation y = x^2 + 3x - 2. To find this equation, the derivative of the curve was used to calculate the slope at the given point, and then integration was used to find the equation of the curve. The process of finding the equation of a curve involves finding the derivative, calculating the slope at a given point, and using integration to find the equation.
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lude1
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Homework Statement



A curve has slope 2x + 3 at each point (x, y) on the curve. Which of the following is an equation for this curve if it pases through the point (1,2)?

Answer: y= x2 + 3x - 2

Homework Equations



y= mx + b

The Attempt at a Solution



If the slope is 2x + 3, then that means it is the derivative. Since I a point (1,2), I can find m and b.

m = 2(1) + 3 = 5​

Thus,

2 = 5(1) + b
b = -3​

Therefore, my equation is

y= 5x - 3​

But this isn't correct. What am I doing wrong?
 
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  • #2
You found the equation of the tangent to the curve at (1,2), not the equation of the curve.
 
  • #3
vela said:
You found the equation of the tangent to the curve at (1,2), not the equation of the curve.

i agree.
To find the equation of the curve you will be doing some integration.
 

1. What is the equation of a curve passing through (1,2)?

The equation of a curve passing through (1,2) can vary depending on the type of curve. In general, it can be written in the form y = mx + b, where m is the slope and b is the y-intercept. However, for more complex curves, the equation may involve multiple variables and exponents.

2. How do I find the equation of a curve passing through (1,2)?

To find the equation of a curve passing through (1,2), you will need to have at least one other point on the curve. Once you have another point, you can use the slope formula (m = (y2 - y1) / (x2 - x1)) to find the slope. Then, plug in the slope and the coordinates of (1,2) into the equation y = mx + b and solve for b. This will give you the equation of the curve passing through (1,2).

3. Can there be multiple equations for a curve passing through (1,2)?

Yes, there can be multiple equations for a curve passing through (1,2). This is because there are infinite possible curves that can pass through a single point. However, if you have additional points on the curve, you can use them to narrow down the possible equations to find the specific curve passing through (1,2).

4. How do I graph the equation of a curve passing through (1,2)?

To graph the equation of a curve passing through (1,2), you can use the slope-intercept form (y = mx + b) to plot the y-intercept (b) and then use the slope (m) to find additional points on the curve. You can also use a graphing calculator or online graphing tool to plot the equation and see the curve passing through (1,2).

5. Can the equation of a curve passing through (1,2) be used to predict other points on the curve?

Yes, the equation of a curve passing through (1,2) can be used to predict other points on the curve. By plugging in different values for x into the equation, you can find the corresponding y-values and plot them on a graph. This can help you visualize the curve and make predictions about its shape and behavior.

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