Function g is a transformation of the parent function f(x)=x^2. The graph of g is a translation left 4 units and down 2 units of the graph of f. Write the equation for h in the form y=ax^2+bx+c.

The answer is A. a translation right 2 units

I just took the assignment. see attached Picture

The answer is A. a translation right 2 units

I just took the assignment. see attached Picture

1. The vertex form of a quadratic equation is f(x) = a(x - h)² + k where (h, k) is the parabola formed by the equation. 2. The value of a affects the shape of the parabola. Three concrete ways showing this are: i. When a is negative ( a < 0), then the parabola opens downward ii. When a is positive ( a > 0), the the parabola opens upward ii. Lastly, when a reduces, the parabola shrinks. And when a increases, the parabola expands as well. 3. Since h directly affects the value of x, then it means that when h is increased by one unit, the parabola moves to the left by one unit. Similarly, if h decreases by 1 unit, it shifts to the right by one unit. Some textbooks call this as the parabola's horizontal shift. 4. The value of k directly affects the movement of the parabola across the y-axis. That means, if k is increased, the parabola goes up. And when k decreases, the graph goes down as well. 5. We have f(x) = 1(x)² as the original function with (h, k) = (0, 0). If we reflect it, across the x-axis, that means we negate the value across. So, we now have a new function, g(x), g(x) = -(x)². Based from the discussion regarding translations, if we move f(x) 5 units to the left, that means we are to increase the value of h by 5. So now, g(x) becomes g(x) = -(x - 5)² Applying the same concept, if we shift the graph 1 unit below, we decrease the value of k by 1. So we now have a final function of g(x) = -(x - 5)² - 1 6. Using the same initial function with 5, we have f(x) = 1(x)² with (h, k) = (0, 0). Now, since f(x) is to be compressed by 3, g(x) becomes g(x) = 1/3(x)² Translating 4 units to the right means decreasing the value of h by 4 and translating 2 units upwards means increasing the value of k by 2. Thus, we have g(x) = 1/3[x - (-4)]² + 2 Simplifying this, we'll have the new function as g(x) = 1/3(x + 4)² + 2

1. The vertex form of a quadratic equation is f(x) = a(x - h)² + k where (h, k) is the parabola formed by the equation. 2. The value of a affects the shape of the parabola. Three concrete ways showing this are: i. When a is negative ( a < 0), then the parabola opens downward ii. When a is positive ( a > 0), the the parabola opens upward ii. Lastly, when a reduces, the parabola shrinks. And when a increases, the parabola expands as well. 3. Since h directly affects the value of x, then it means that when h is increased by one unit, the parabola moves to the left by one unit. Similarly, if h decreases by 1 unit, it shifts to the right by one unit. Some textbooks call this as the parabola's horizontal shift. 4. The value of k directly affects the movement of the parabola across the y-axis. That means, if k is increased, the parabola goes up. And when k decreases, the graph goes down as well. 5. We have f(x) = 1(x)² as the original function with (h, k) = (0, 0). If we reflect it, across the x-axis, that means we negate the value across. So, we now have a new function, g(x), g(x) = -(x)². Based from the discussion regarding translations, if we move f(x) 5 units to the left, that means we are to increase the value of h by 5. So now, g(x) becomes g(x) = -(x - 5)² Applying the same concept, if we shift the graph 1 unit below, we decrease the value of k by 1. So we now have a final function of g(x) = -(x - 5)² - 1 6. Using the same initial function with 5, we have f(x) = 1(x)² with (h, k) = (0, 0). Now, since f(x) is to be compressed by 3, g(x) becomes g(x) = 1/3(x)² Translating 4 units to the right means decreasing the value of h by 4 and translating 2 units upwards means increasing the value of k by 2. Thus, we have g(x) = 1/3[x - (-4)]² + 2 Simplifying this, we'll have the new function as g(x) = 1/3(x + 4)² + 2

BBCB
(x-1) would be the opposite on the x axis so it will be 1 to the right.
if it is (x)-1 it is accurate to the horizontal change so it is 1 down

Problem 2

B) Shift 9 units to the left

If we replace every x with x+9, then the inputs are now 9 units larger. This moves the xy axis 9 spaces to the right. If we keep the f(x) curve fixed in place while the xy axis moves, then it gives the illusion f(x) moves 9 units to the left.

Problem 3

B) Shift 1 unit down

Think of f(x) as y. So f(x)-1 means y-1 to show that we subtract 1 from each y coordinate of each (x,y) point on f(x).

Problem 4

C) g(x) = 2f(x)

We multiply each y value by 2. So g(x) = 2y = 2*f(x). This makes f(x) twice as tall as before.

Problem 5

B) shifted to the right 3 units and up 1 unit

The replacement of x with x-3 means we shift 3 units to the right. It's the similar idea to problem 2, but we're working in reverse now.

The +1 at the end means we shift 1 unit up.

Problem 2

B) Shift 9 units to the left

If we replace every x with x+9, then the inputs are now 9 units larger. This moves the xy axis 9 spaces to the right. If we keep the f(x) curve fixed in place while the xy axis moves, then it gives the illusion f(x) moves 9 units to the left.

Problem 3

B) Shift 1 unit down

Think of f(x) as y. So f(x)-1 means y-1 to show that we subtract 1 from each y coordinate of each (x,y) point on f(x).

Problem 4

C) g(x) = 2f(x)

We multiply each y value by 2. So g(x) = 2y = 2*f(x). This makes f(x) twice as tall as before.

Problem 5

B) shifted to the right 3 units and up 1 unit

The replacement of x with x-3 means we shift 3 units to the right. It's the similar idea to problem 2, but we're working in reverse now.

The +1 at the end means we shift 1 unit up.

BBCB
(x-1) would be the opposite on the x axis so it will be 1 to the right.
if it is (x)-1 it is accurate to the horizontal change so it is 1 down

To solve the given task, let's break it down step by step.

10. Describing the transformation of the equation from the parent function y = |x| to y = |x - 4|:
A. Down 4
B. Up 4
C. Right 4
D. Left 4

The correct answer is C. Right 4.

Explanation: The transformation of the parent function y = |x| is achieved by shifting the graph 4 units to the right. The equation y = |x - 4| reflects this transformation by subtracting 4 from the x-coordinate of the absolute value function.

11. Identifying the type of function shown:
A. Square root
B. Absolute value
C. Linear
D. Quadratic

The correct answer is B. Absolute value.

Explanation: The equation y = |x - 4| represents an absolute value function. The absolute value function is characterized by the "V" shape in its graph.

12. Describing the transformation applied to the parent function f(x) = |x| to obtain f(x) = |x - 5|:
A. Shifts right 5
B. Shifts left 5
C. Shifts down 5
D. Shifts up 5

The correct answer is A. Shifts right 5.

Explanation: The transformation of the parent function f(x) = |x| is achieved by shifting the graph 5 units to the right. The equation f(x) = |x - 5| reflects this transformation by subtracting 5 from the x-coordinate of the absolute value function.

13. Choosing the correct translated function:
A. f(x) = |x| + 2
B. f(x) = -|x|
C. f(x) = |x| - 2
D. f(x) = -|x| + 2

The correct answer is D. f(x) = -|x| + 2.

Explanation: Among the given options, the function f(x) = -|x| + 2 represents the correct translation. The negative sign outside the absolute value reflects a reflection of the graph across the x-axis, and the "+2" term shifts the graph 2 units upward.

Please note that the provided explanations are based on the information given in the task. If any additional information is provided, the solutions might change accordingly.

See explanation

Step-by-step explanation:

1. The function represents the amount of profit Bill makes per toy when he increases or decreases the price of his handmade toys.

The graph of this function is a parabola. The maximum profit is at the vertex of the parabola.

Find the vertex:

This means that decrease of 1 money unit (dollar, cent, euro,...) will give Bill the maximum profit of 4 money units (dollars, cents, euros,...)

2. The height, in feet, of the pebble over time is modeled by the equation

The maximum height of the pebble is at parabola's vertex.

Find the vertex:

Thus, the maximum height the pebble can reach is 96 feet.

3. First, reflect across the x-axis the graph of the parent function The reflection across the x-axis will give us the function

Now, translate this function left 4 units and down 2 units to form the graph of g(x):

Open the brackets:

Hence,

4. The shape of the inside of a glass follows a parabola with the function

The vertex of the parabola represents the bottom of the inside of the glass. Find the vertex of the parabola:

So, point (-3,0) represents the bottom of the inside of the glass.

5. The equation y = −4.9x2 + 14 represents the height y in meters of a rock dropped off a bridge over time x in seconds.

The graph of the function is attached.

6. First, reflect across the x-axis the graph of the parent function The reflection across the x-axis will give us the function

Now, translate this function right 5  units and up 1 unit to form the graph of g(x):

Open the brackets:

Hence,