What are the remaining steps to complete this problem?

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Homework Help Overview

The problem involves finding various properties of the quadratic function y = x² + 3x - 4, including tangent and normal lines at specific points, local extrema, and areas related to these tangents. The original poster seeks clarification on parts d and e of the problem.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the conditions for finding tangent lines from a point not on the curve and the implications of the quadratic nature of the function. There is exploration of the relationship between the slopes of the tangent and the line segment connecting a point on the curve to the external point (2, -4). Questions arise about the interpretation of the area calculation in part e and the definition of "regular slope."

Discussion Status

Some participants have provided insights into the geometric interpretation of the problem, particularly regarding the conditions for tangency and the calculation of areas. There is ongoing clarification about the requirements of parts d and e, with no explicit consensus reached on the interpretations yet.

Contextual Notes

There is uncertainty regarding the exact wording of the problem, particularly in part e, which may affect the understanding of the area calculation. The original poster's confusion about the terminology used in the problem is also noted.

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Homework Statement


Given the function
y = x^{2} + 3x - 4 find,

a) the equation of tangent + normal at x = -2.
b)the equation of the tangent that has slope of -2.
c)local min and max.
d)the equations of the tangent from the point (2 , -4) to the curve.
e) area enclosed by the tangent at x = -2 and the axis.



The Attempt at a Solution


ok so i did parts a to c. I am kind of stuck at part d and e. I don't understand what the questions mean.
 
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d) Notice that (2, -4) is not on the curve. We want the equations of all lines passing through (2, -4) that are tangent to the curve. There could be 0, 1, or 2 of these lines because y is a quadratic. Let (x, y) be a point on one such line that's also on the curve. Then you know what y is in terms of x, and you can find the slope of the line in 2 ways: by differentiation, and by rise over run from (x, y) to (2, -4).

e) You found the equation of the tangent at x = -2 in part (a). Graph it and calculate the area of the triangle enclosed by this line, the x-axis, and the y-axis.
 
Part d is a bit tricky, I think. You want to find a point A on the parabola with coordinates (p, p^2 + 3p - 4) that satisifies these conditions:
  1. The slope of the segment from A to (2, -4) is equal to the slope of the tangent at A (p, p^2 + 3p -4).
  2. There is only one value of p that works.
I haven't worked through this, but this is the approach I would take for starters. The reason for the second condition is that the tangent has to touch the graph of the parabola at exactly one point; it can't intersect the graph at a point and then continue to intersect the other side of the parabola graph.

Part e is not well-defined. Did the original problem say "area enclosed by the tangent at x = -2 and the axes"? One axis, two axes. If so, the area involved is a triangle, so if you can find the intercepts of the tangent on the two axes, it's pretty easy to find the area of that triangle.
 
I don't understand why the tangent slope is equal to the regular slope.
 
I'm not sure what you're asking. What do you mean by "regular slope"?
 
sorry. I was referring to the slope from point A to point (2 , 4)
 
Your problem asks for "the equations of the tangent from the point (2 , -4) to the curve."
I interpret this to mean the line that is tangent to the given curve at A(p, p^2 + 3p + 4)and that passes through (2, -4). Can you find the slope of this line segment? Can you find the slope of the tangent line to the curve at point A? The two expressions you get will involve p and have to be equal.
 

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