Question16) If $f(x)=2-(x-1)^{3}$ , then the graph that represents the function $f$ is (a) (b) (c) (d) 17) The rule of the function represented in the opposite figure is $f(x)=$ $\qquad$ (a) $(x-2)^{2}+1$ (b) $-(x-2)^{2}+1$ (c) $-(x-1)^{2}+2$ (d) $(-x+1)^{2}+2$ 18) The symmetric point of the function $f: f(x)=x^{3}-2$ is $\qquad$ (a) $(0,2)$ (b) $(0,-2)$ (c) $(2,0)$ (d) $(-2,0)$ 19) The vertex of the curve of the function $f$ where $f(x)=(1+x)^{2}-3$ is $\qquad$ (a) $(1,3)$ (b) $(1,-3)$ (c) $(-1,3)$ (d) $(-1,-3)$ 20) If $y=f(x)$ is a real function, then its image by translation 3 units vertically upwards is $g(x)=$ $\qquad$ (a) $f(x-3)$ (b) $f(x+3)$ (c) $f(x)+3$ (d) $(x)-3$

Studdy Solution

STEP 1

1. We need to identify the graph of the function $f(x) = 2 - (x-1)^3$ .

STEP 2

1. Analyze the transformation of the basic cubic function.
2. Determine the characteristics of the graph.
3. 1. Analyze the vertex form of the parabola.
2. Match the given options with the characteristics of the parabola.
STEP_1: High_Level_Step: 1 The vertex form of a parabola is $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex.
High_Level_Step_Completed: FALSE
STEP_2: High_Level_Step: 1 Identify the vertex and the direction of the parabola from the figure.
High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 2 Match the vertex and direction with the options provided.
High_Level_Step_Completed: TRUE
Problem 18:
_ASSUMPTIONS_:
1. We need to find the symmetric point of the function $f(x) = x^3 - 2$ .
_HIGH_LEVEL_APPROACH_:
1. Understand symmetry in cubic functions.
2. Determine the symmetric point.
STEP_1: High_Level_Step: 1 Cubic functions like $f(x) = x^3$ are symmetric about the origin.
High_Level_Step_Completed: FALSE
STEP_2: High_Level_Step: 2 Since $f(x) = x^3 - 2$ , the graph is shifted down by 2 units, affecting the symmetry.
High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 2 The symmetric point relative to the origin shift is $(0, -2)$ .
High_Level_Step_Completed: TRUE
Problem 19:
_ASSUMPTIONS_:
1. We need to find the vertex of the function $f(x) = (1+x)^2 - 3$ .
_HIGH_LEVEL_APPROACH_:
1. Identify the vertex form of the quadratic function.
2. Determine the vertex coordinates.
STEP_1: High_Level_Step: 1 The function is in the form $f(x) = (x-h)^2 + k$ .
High_Level_Step_Completed: FALSE
STEP_2: High_Level_Step: 2 Here, $h = -1$ and $k = -3$ , so the vertex is $(-1, -3)$ .
High_Level_Step_Completed: TRUE
Problem 20:
_ASSUMPTIONS_:
1. We need to find the new function after a vertical translation.
_HIGH_LEVEL_APPROACH_:
1. Understand vertical translation of functions.
2. Apply the translation to the function.
STEP_1: High_Level_Step: 1 Vertical translation upwards by 3 units adds 3 to the function: $g(x) = f(x) + 3$ .
High_Level_Step_Completed: TRUE
The answers to the problems are: 16) The graph is a cubic curve shifted right by 1, reflected, and shifted up by 2. 17) Match the vertex and direction with the options. 18) The symmetric point is $(0, -2)$ . 19) The vertex is $(-1, -3)$ . 20) The translated function is $g(x) = f(x) + 3$ .

STEP 3

The function $f(x) = 2 - (x-1)^3$ is a transformation of the basic cubic function $y = x^3$ .

STEP 4

The transformation involves: - A horizontal shift to the right by 1 unit, due to $(x-1)$ . - A reflection across the horizontal axis, due to the negative sign in front of $(x-1)^3$ . - A vertical shift upwards by 2 units, due to the "+2".

STEP 5

The graph will be a cubic curve that is shifted right by 1 unit, reflected over the x-axis, and shifted up by 2 units.
Problem 17:
_ASSUMPTIONS_:
1. We need to find the equation of a parabola given in a figure.

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