What is the vertex of the graph of f(x) = |x + 3| + 7?
(3, 7), (7, 3), (-3, 7), (7, -3)

Solution:

Given: f(x) = |x+3|+7

To find: The vertex of the graph

We know that the vertex of the graph is the point of intersection of the two lines given.

The equation f(x) = ǀx +3ǀ + 7 actually results in two equations because of the absolute function ǀx +3ǀ which generates two scenarios:

Scenario1: When x +3 > 0

y = f(x) = x + 3 + 7 = x +10 --- (1)

Scenario 2: When x + 3 < 0 ǀx + 3ǀ = -(x + 3)

y = f(x) = - x - 3 + 7 = -x + 4 --- (2)

Solving linear equations (1) and (2) simultaneously we have

y =   x + 10

y = - x + 4

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2y = 0 + 14

y = 14/2 = 7--- (3)

Substituting (3) in either (1) or (2) we get

x = -3

The vertex of the graph is (-3, 7).

The graph of f(x) = |x + 3| + 7 comprises two lines intersecting with each other and forming a triangle with the help of the x axis (y=0).

The point of intersection of the two lines (represented by the linear equations y = x + 10 and y = -x + 4) is the coordinate (-3, 7) which is the solution to the problem.

What is the vertex of the graph of f(x) = |x + 3| + 7?

Summary:

The vertex of the graph of f(x) = |x + 3| + 7 is (-3, 7).