The figure below shows a partial graph of two functions, f(x)f(x) and g(x)g(x). f(x)\f(x) is a parabola defined by the quadratic function f(x)=-(x+2)^{2}+17f(x)=−(x+2)
2
+17 and g(x)g(x) is a linear function that passes through the points \left(0,-1\right)(0,−1) and (1,0)(1,0).

Respond to Part A and Part B based on the given information.

Part A
Determine the x-x−coordinate of the point in Quadrant III where \ f(x)=g(x) f(x)=g(x).

Part B
Justify algebraically why the xx-coordinate you identified in Part A is a solution to the equation f(x)=g(x).f(x)=g(x).

Question

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Respuesta :

MrRoyal MrRoyal · Answer 1 · 2022-06-11T18:59:52+02:00

The x-coordinate of the point in quadrant III where f(x) = g(x) is -7

The figure is not attached to the question; however, the question can still be solved

The quadratic function is given as:

f(x) = -(x +2)² + 17

The linear function passes through (0,-1) and (1,0).

So, we calculate the linear equation using:

[tex]y = \frac{y_2 - y_1}{x_2 -x_1} * (x -x_1) + y_1[/tex]

Substitute known values

[tex]y = \frac{0+ 1}{1-0} * (x -0) -1[/tex]

Evaluate

y = x - 1

So, we have:

Next, we plot the graph of both functions (see attachment)

From the attached graph, both functions meet in the quadrant III at (-7,-8)

Hence, the x-coordinate of the point in quadrant III where f(x) = g(x) is -7

In (a), we have the solution to be:

(x,y) = (-7,-8)

Substitute -7 for x in f(x) and g(x)

f(-7) = -(-7 +2)² + 17 = -8

g(-7) = -7 - 1 = -8

See that:

f(-7) = g(-7) = -8 and it is located in the third quadrant.

Hence, the solution in (a) is true