-gre.ge.04 intersection of parabola and line

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Discussion Overview

The discussion revolves around finding the intersection points of a linear function and a quadratic function in the xy-plane, specifically focusing on determining the value of v, the x-coordinate of one of the intersection points. The context includes mathematical reasoning and exploration of the relationships between the functions involved.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant notes that the value of v cannot be determined solely by observation from the graph, as it is not to scale, and suggests proceeding with the equations of the functions.
  • Another participant provides the equation of the line as \(y=4x-9\) and mentions the secant line from (0,3) to (4,19) being parallel to this line.
  • A participant derives the quadratic equation from the intersection conditions, leading to \(x^2 - 4x - 12 = 0\) and factors it to find \(v=6\).
  • There is a correction regarding the coordinates of a point, emphasizing that if \(x=2\), the corresponding y-value does not match the expected output, indicating a misunderstanding of the graph's representation.
  • Another participant suggests that the coordinates for \((v,w)\) could be limited to two possibilities, \((5,11)\) and \((6,15)\), based on the graphical representation and the relationships between the points.

Areas of Agreement / Disagreement

Participants express differing views on the determination of the intersection points, with some proposing specific values for v while others highlight the uncertainty in the graphical representation. No consensus is reached on the exact coordinates of the intersection points.

Contextual Notes

There are limitations regarding the accuracy of the graphical representation, which affects the ability to ascertain the coordinates of the intersection points. Additionally, assumptions about the behavior of the functions and their intersections are not fully resolved.

karush
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$\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function.
The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$,
what is the value of $\textbf{v}$?
${-6}\quad {6}\quad {5}\quad {7}\quad {8}$

ok before I plow into this one it seems obvious that v could not be known for certain by observation
(the graph does not look it is to scale)
so then we can only proceed with the intersections of the equations of
$$y=a(x-4)^2 +19 \quad y=\dfrac{9}{2}x-9$$

unless some other quickie could apply
 
Last edited:
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the line has equation $y=4x-9$

note the secant line from (0,3) to (4,19) is parallel to the line $y=4x-9$
 
skeeter said:
the line has equation $y=4x-9$
note the secant line from (0,3) to (4,19) is parallel to the line $y=4x-9$
$-\left(x^{2}-8x+16\right)-4x+9+19=0$
$x^{2}-4x-12=0$
$(x-6)(x+2)$
v=6

ok I couldn't see how the secant would make things obvious
 
Last edited:
karush said:

$\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function.
The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$,
what is the value of $\textbf{v}$?
${-6}\quad {6}\quad {5}\quad {7}\quad {8}$

ok before I plow into this one it seems obvious that v could not be known for certain by observation
(the graph does not look it is to scale)
so then we can only proceed with the intersections of the equations of
$$y=a(x-4)^2 +19 \quad y=\dfrac{9}{2}x-9$$
No! If x= 2, this gives y= 9- 9= 0, not -1. The point (2, -1) is just below the x-axis, not on it.

unless some other quickie could apply
 
karush said:
$-\left(x^{2}-8x+16\right)-4x+9+19=0$
$x^{2}-4x-12=0$
$(x-6)(x+2)$
v=6

ok I couldn't see how the secant would make things obvious

$\dfrac{w - (-1)}{v - 2} = 4$

note from the graph that $4 <v < 8$ and $3 < w < 19$

so, only two possible coordinates for $(v,w)$ ...

$(5,11)$ and $(6, 15)$

$(5,11)$ would be vertically midway between $(0,3)$ and $(4,19)$ if it were $(v,w)$.
 
ok i see
mahalo much
 

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