Let's break down each option to determine the correct statement regarding the parent function of the given equations and their graph shapes.
1. The equation [tex]\( f(x) = 1.4|x-1| \)[/tex] is an absolute value function and has a [tex]\( V \)[/tex]-shaped graph.
- The function [tex]\( f(x) = 1.4|x-1| \)[/tex] is indeed an absolute value function.
- The graph of this function takes the form of a [tex]\( V \)[/tex]-shape, centered at [tex]\( x = 1 \)[/tex] and scaled vertically by a factor of 1.4.
This statement is correct.
2. The equation [tex]\( f(x) = -3x^2 - 9 \)[/tex] is a quadratic function and has a vertical [tex]\( S \)[/tex]-shaped graph.
- The function [tex]\( f(x) = -3x^2 - 9 \)[/tex] is indeed a quadratic function.
- However, the graph of this function is a parabola that opens downwards, not an [tex]\( S \)[/tex]-shape. A quadratic function has a [tex]\( U \)[/tex]-shaped or an inverted [tex]\( U \)[/tex]-shaped graph, depending on the sign of the leading coefficient.
This statement is incorrect.
3. The equation [tex]\( f(x) = 8 \sqrt{x} - 3.5 \)[/tex] is a cube root function and has a graph that is half of a sideways parabola.
- The function [tex]\( f(x) = 8 \sqrt{x} - 3.5 \)[/tex] is a square root function, not a cube root function.
- The graph of this function is indeed half of a sideways parabola, but since it is identified incorrectly as a cube root function, the statement is flawed.
This statement is incorrect.
4. The equation [tex]\( f(x) = 4 \tan x + 1 \)[/tex] is a tangent function and has a graph consisting of waves that go up and down at regular intervals.
- The function [tex]\( f(x) = 4 \tan x + 1 \)[/tex] is indeed a tangent function.
- The graph of the tangent function has repeating waves that go up and down at regular intervals, with vertical asymptotes separating each wave.
This statement is correct.
Considering the analysis, the correct choices are:
1. The equation [tex]\( f(x) = 1.4|x-1| \)[/tex] is an absolute value function and has a [tex]\( V \)[/tex]-shaped graph.
2. The equation [tex]\( f(x) = 4 \tan x + 1 \)[/tex] is a tangent function and has a graph consisting of waves that go up and down at regular intervals.
However, since only one correct choice was requested to be identified in the problem, the final answer is:
The equation [tex]\( f(x) = 1.4|x-1| \)[/tex] is an absolute value function and has a [tex]\( V \)[/tex]-shaped graph.
