The graph of y=x^2 is reflected in the x-axis, then stretched vertically by a factor of 2, and then translated 3 units to the left and 1 unit down. I need to write the equation of this parabola in standard and general form. I have 4 other questions so could someone please show me all the steps involved so I can learn to do them. Please and thank you so much.

start with

y = x^2

"reflected in the x-axis" ---> y = -x^2
"stretched vertically by a factor of 2 ---> y = -2x^2
"translated 3 units to the left, 1 unit down
----> y = -2(x+3) -1

Oh, it's time to put on my math hat and get serious! But don't worry, I'll still sprinkle in a little humor here and there to keep things lively. Let's start crunching those numbers!

1. Reflection in the x-axis:
When a graph is reflected in the x-axis, the sign of the y-coordinate is flipped. So, our new equation becomes y = -x^2.

2. Vertical stretching by a factor of 2:
Stretching vertically by a factor of 2 means each y-coordinate is multiplied by 2. Since we already have y = -x^2, our new equation is y = -2x^2.

3. Translation 3 units to the left and 1 unit down:
To translate the graph 3 units to the left, we replace x with (x + 3) in our equation. And to move it 1 unit down, we subtract 1 from the y-coordinate. After these translations, our equation becomes y = -2(x + 3)^2 - 1.

Now let's convert this equation to standard form:

1. Expand and simplify:
y = -2(x + 3)^2 - 1
= -2(x^2 + 6x + 9) - 1
= -2x^2 - 12x - 18 - 1
= -2x^2 - 12x - 19

So, the equation of the parabola after reflection, vertical stretching, and translation is:
Standard form: y = -2x^2 - 12x - 19
General form: 2x^2 + 12x + (y + 19) = 0 (If you want the fun and general form)

I hope my bot-brain calculations were helpful! Let me know if you have any other questions or if you want me to add a bit more humor to these equations. Maths can be fun too, you know!

Sure! Let's break down the given transformations step-by-step to find the equation of the transformed parabola.

Step 1: Reflecting in the x-axis
When a graph is reflected in the x-axis, the sign of the y-coordinate is flipped. So the vertex (the lowest point) of the parabola remains the same, but all the other points are reflected on the opposite side of the x-axis.

Step 2: Stretching vertically by a factor of 2
When a graph is stretched vertically by a factor of k, all the y-coordinates are multiplied by the factor k. In this case, the graph is stretched vertically by a factor of 2, so the y-coordinate of each point is multiplied by 2.

Step 3: Translating 3 units to the left and 1 unit down
When a graph is translated to the left or right, the x-coordinate of each point is shifted by the specified amount. In this case, the graph is translated 3 units to the left, so we subtract 3 from the x-coordinate of each point. Additionally, it is translated 1 unit down, so we subtract 1 from the y-coordinate of each point.

Let's go through these steps one by one to find the equation of the transformed parabola.

Step 1: Reflecting in the x-axis
The original equation of the parabola is y = x^2. When we reflect it in the x-axis, the y-coordinate becomes negative. Therefore, the equation is now y = -x^2.

Step 2: Stretching vertically by a factor of 2
The equation after the vertical stretch by a factor of 2 will be y = -2x^2.

Step 3: Translating 3 units to the left and 1 unit down
To translate the graph 3 units to the left and 1 unit down, we need to subtract 3 from the x-coordinate and subtract 1 from the y-coordinate. The resulting equation will be:

y = -2(x+3)^2 - 1

This is the equation of the transformed parabola in vertex form.

To write the equation in general form, we can expand and simplify the equation:

y = -2(x+3)^2 - 1
y = -2(x^2 + 6x + 9) - 1
y = -2x^2 - 12x - 18 - 1
y = -2x^2 - 12x - 19

So, the equation of the transformed parabola in general form is y = -2x^2 - 12x - 19.

I hope this helps! Let me know if you have any further questions.

Sure, I'd be happy to help you with this and show you the steps involved. Let's go through the process step by step to find the equation of the transformed parabola.

Step 1: Start with the equation of the original parabola, y = x^2.

Step 2: Reflect the graph in the x-axis. This means that every y-coordinate of the points on the graph becomes its opposite. So, the equation becomes y = -x^2.

Step 3: Stretch the graph vertically by a factor of 2. This means that we multiply each y-coordinate by 2. Therefore, the equation becomes y = -2x^2.

Step 4: Translate the graph 3 units to the left and 1 unit down. To translate the graph 3 units to the left, we replace x with (x + 3). And to translate it 1 unit down, we subtract 1, giving us (x + 3)^2 - 1. Therefore, the equation becomes y = -2(x + 3)^2 - 1.

Now, let's rewrite this equation in both standard and general form.

Standard form:
The standard form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants.

In this case, the equation is y = -2(x + 3)^2 - 1. To expand the equation, we apply the distributive property: y = -2(x^2 + 6x + 9) - 1.

Next, we multiply each term inside the brackets by -2 to get: y = -2x^2 - 12x - 18 - 1.

Simplifying further, we combine like terms: y = -2x^2 - 12x - 19. Therefore, the equation in standard form is y = -2x^2 - 12x - 19.

General form:
The general form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants, and it can be written without expanding.

Using the equation y = -2(x + 3)^2 - 1, we can further simplify it by expanding the squared term: y = -2(x^2 + 6x + 9) - 1.

Applying the distributive property, we get: y = -2x^2 - 12x - 18 - 1.

Combining like terms, we have: y = -2x^2 - 12x - 19. Therefore, the equation in general form is y = -2x^2 - 12x - 19.

I hope this explanation helps you understand how to transform the equation of a parabola and write it in standard and general form. If you have any further questions, please let me know!