-gre.ge.04 intersection of parabola and line

In summary, the xy-plane above shows one of the two points of intersection of the graphs of a linear function and a quadratic function. The shown point of intersection has coordinates (v,w). If the vertex of the graph of the quadratic function is at (4,19), the value of v could be either 5 or 6 based on the given constraints.
  • #1
karush
Gold Member
MHB
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054.png

$\textbf{xy-plane}$ 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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  • #2
the line has equation $y=4x-9$

note the secant line from (0,3) to (4,19) is parallel to the line $y=4x-9$
 
  • #3
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:
  • #4
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
 
  • #5
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 $(v,w)$.
 
  • #6
ok i see
mahalo much
 

Related to -gre.ge.04 intersection of parabola and line

1. What is the intersection of a parabola and a line?

The intersection of a parabola and a line is the point or points where the two curves intersect. This point can be found by solving the equations of the parabola and the line simultaneously.

2. How can I find the coordinates of the intersection?

To find the coordinates of the intersection, you can set the equations of the parabola and the line equal to each other and solve for the variables. The resulting values will be the x and y coordinates of the intersection point.

3. Can a parabola and a line intersect at more than one point?

Yes, a parabola and a line can intersect at more than one point. This occurs when the parabola and the line have more than one point in common, which can happen if the line is tangent to the parabola or if the parabola is a higher degree curve.

4. What does the intersection of a parabola and a line represent?

The intersection of a parabola and a line represents the solution to the system of equations formed by the two curves. It is the point where the x and y values satisfy both equations simultaneously.

5. Is there a specific method for finding the intersection of a parabola and a line?

Yes, there are several methods for finding the intersection of a parabola and a line. These include graphing the two curves and visually determining the point of intersection, using substitution or elimination to solve the system of equations, or using calculus to find the derivative of the parabola and setting it equal to the slope of the line at the point of intersection.

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