Graph function using the techinique of shifting, compressing, strecthing and /or reflecting

f(x)=5(x+4)^2+1

graph x^2

shift left by 4: (x+4)^2
stretch 5: 5(x+4)^2
shift up 1: 5(x+4)^2+1

To graph the function f(x)=5(x+4)^2+1 using the techniques of shifting, compressing, stretching, and/or reflecting, we need to understand how each of these operations affects the graph.

1. Shifting: The function is of the form f(x+a), where 'a' indicates the horizontal shift. If 'a' is positive, the graph moves left, and if 'a' is negative, the graph moves right.

In this case, f(x)=5(x+4)^2+1, we have a horizontal shift of -4 units to the left since x+4 is equivalent to (x+(-4)). Therefore, the graph is shifted 4 units to the left.

2. Compressing or Stretching: The term inside the square, (x+4), affects the shape and size of the parabola. If the term is multiplied by a positive number greater than 1, it compresses the graph horizontally, making it narrower. Conversely, if the term is multiplied by a positive number between 0 and 1, it stretches the graph horizontally, making it wider.

In our function, (x+4) is not multiplied or divided by any value. So, the graph is neither compressed nor stretched horizontally.

3. Reflecting: The function is of the form f(-x), where replacing 'x' with '-x' reflects the graph across the y-axis. This means that all points on the graph will have their x-coordinate negated.

In this case, f(x)=5(x+4)^2+1, we do not have a reflection since there is no negative sign in front of 'x'.

Now that we understand the effects of shifting, compressing, stretching, and reflecting, we can graph the function step by step:

1. Start with a rough sketch of the parent function, in this case, f(x)=x^2.
2. Apply the horizontal shift of -4 units to the left. Each point on the graph will move 4 units to the left.
3. Multiply the function by 5, which will scale the graph vertically. Each y-coordinate is multiplied by 5.
4. Translate the graph upward by 1 unit, adding 1 to each y-coordinate.

The final graph will be a parabola shifted 4 units to the left, vertically stretched by a factor of 5, and translated upward by 1 unit.