Compare each function with the parent function. Without graphing what are the vertex, axis of symmetry, and transformations of the parent function?
Y=-|8x+4|+2
A) (1/2,2);x=-1/2; translated to the left 1/2 unit and up 2 units
B) (-1/2,2);x=-1/2; translated to the left 1/2 unit, up 2 units, and reflected in the x-axis
C) (1/2,2);x=1/2; translated to the right 1/2 unit up and 2 units, and reflected in the y-axis
D) (1/2,-2); x=1/2; translated to the right 1/2 unit and up 2 units
To find the vertex, axis of symmetry, and transformations of the parent function, we need to identify the changes made to the parent function.
The parent function of this equation is y = |x|. The transformations applied to this parent function are as follows:
1. Vertical Translation: The equation is y = -|8x + 4| + 2. The "+2" term at the end translates the graph upward by 2 units.
2. Horizontal Translation: The equation is still y = -|8x + 4| + 2. The "-4" term inside the absolute value translates the graph to the left by 4 units.
3. Reflection: There is no reflection in the x-axis because there is no negative sign in front of the absolute value. Therefore, there is no reflection in the y-axis either.
Now, let's compare the given options with the identified transformations to find the correct answer.
A) (1/2, 2); x = -1/2; translated to the left 1/2 unit and up 2 units.
This option does not match the identified horizontal translation of the equation or the vertex coordinates.
B) (-1/2, 2); x = -1/2; translated to the left 1/2 unit, up 2 units, and reflected in the x-axis.
This option matches the identified horizontal translation and vertical translation of the equation. However, there is no reflection in the x-axis, so this option is not correct.
C) (1/2, 2); x = 1/2; translated to the right 1/2 unit and up 2 units, and reflected in the y-axis.
This option does not match the identified horizontal translation or vertex coordinates. Additionally, there is no reflection in the y-axis.
D) (1/2, -2); x = 1/2; translated to the right 1/2 unit and up 2 units.
This option matches the identified horizontal translation and vertical translation of the equation. The vertex coordinates also match.
Therefore, the correct answer is D) (1/2, -2); x = 1/2; translated to the right 1/2 unit and up 2 units.
To find the vertex, axis of symmetry, and transformations of the parent function, you need to understand the general form of the function and how each term affects the graph.
The parent function of the given function is f(x) = |x|. The parent function is a simple absolute value function with a vertex at (0, 0).
Let's analyze the given function: Y = -|8x + 4| + 2.
1. The function is multiplied by -1, so it is reflected in the x-axis.
2. The "x" inside the absolute value is multiplied by 8, which means it is compressed horizontally by a factor of 1/8.
3. The "x" inside the absolute value is shifted left by 4 units since the equation is "8x + 4". So, we need to reverse that shift by moving the vertex 4 units to the right.
4. Lastly, the entire function is shifted up by 2 units.
Now, let's compare each option to determine which one matches the transformations we just explained:
A) (1/2, 2); x = -1/2; translated to the left 1/2 unit and up 2 units
This option has the correct translations but does not match the axis of symmetry. The axis of symmetry should be x = 4 (reversed shift).
B) (-1/2, 2); x = -1/2; translated to the left 1/2 unit, up 2 units, and reflected in the x-axis
This option has the correct translations and reflects in the x-axis, matching all the given transformations. The vertex and axis of symmetry are also correct.
C) (1/2, 2); x = 1/2; translated to the right 1/2 unit up and 2 units, and reflected in the y-axis
This option does not match the given transformations. The function should be reflected in the x-axis, not the y-axis. The vertex and axis of symmetry are also incorrect.
D) (1/2, -2); x = 1/2; translated to the right 1/2 unit and up 2 units
This option has the correct translations but does not match the vertex. The vertex should be shifted 4 units to the right, not 1/2 unit.
Based on the analysis, the correct option is B) (-1/2, 2); x = -1/2; translated to the left 1/2 unit, up 2 units, and reflected in the x-axis.