The graph of y=x^2-2x+1 is translated by the vector (2 3).The graph so obtained is reflected in the x-axis and finally it is stretched by a factor 2 parallel to y-axis.Find the equation of the final graph is the form y=ax^2+bx+c, where a,b and c are constants to be found.
y = (x-1)^2
To move right 2 and up 3, you get a new
y = ((x-2)-1)^2 + 3 = (x-3)^2 + 3
reflection gives you y = -((x-3)^2 + 3)
stretching gives you y = -2((x-3)^2+3)
Well, doesn't that sound like a wild journey for our graph? Let's break it down step by step, shall we?
First, the graph is translated by the vector (2, 3). This means that we simply add 2 to the x-coordinate of each point and add 3 to the y-coordinate. So, our new equation becomes:
y = (x + 2)^2 - 2(x + 2) + 1
Next, the graph is reflected in the x-axis. This means that we change the sign of the y-coordinate for each point. Hence, our equation becomes:
y = -(x + 2)^2 + 2(x + 2) - 1
Lastly, the graph is stretched by a factor of 2 parallel to the y-axis. This means that we multiply the y-coordinate of each point by 2. Therefore, our final equation becomes:
y = -2(x + 2)^2 + 4(x + 2) - 2
And there you have it! The equation of the final graph, in the form y = ax^2 + bx + c, is:
y = -2x^2 + 4x - 6
Now, I hope this wild ride didn't leave you feeling too stretched out!
To find the equation of the final graph, we need to go through the given transformations step by step.
1. Translation by (2, 3)
The translation of a graph by the vector (a, b) is given by replacing x with (x - a) and y with (y - b). So, for the given graph, we have a translation by (2, 3) which means:
New x = x - 2
New y = y - 3
The equation becomes:
y = (x - 2)^2 - 2(x - 2) + 1
= (x^2 - 4x + 4) - (2x - 4) + 1
= x^2 - 4x + 4 - 2x + 4 + 1
= x^2 - 6x + 9
2. Reflection in the x-axis
The reflection in the x-axis is achieved by multiplying the y-coordinate by -1. So, the equation becomes:
y = -(x^2 - 6x + 9)
3. Stretching by a factor of 2 parallel to the y-axis
To stretch the graph by a factor of 2 parallel to the y-axis, we multiply the entire equation by the square of the stretching factor. In this case, the stretching factor is 2, so the equation becomes:
y = -2(x^2 - 6x + 9)
= -2x^2 + 12x - 18
Finally, we can rewrite the equation in the form y = ax^2 + bx + c:
y = -2x^2 + 12x - 18
So, the equation of the final graph is y = -2x^2 + 12x - 18.
To find the equation of the final graph, we need to follow these steps:
Step 1: Original Equation
Start with the original equation of the graph: y = x^2 - 2x + 1.
Step 2: Translation by Vector (2 3)
To translate the graph by the vector (2 3), we need to add 2 to the x-coordinate and 3 to the y-coordinate of each point on the graph. This can be done by replacing x with (x - 2) and y with (y - 3) in the equation.
So, the translated equation becomes: y = (x - 2)^2 - 2(x - 2) + 1.
Simplifying this equation gives us: y = x^2 - 6x + 9.
Step 3: Reflection in the x-axis
To reflect the graph in the x-axis, we need to replace y with -y in the equation.
So, the reflected equation becomes: -y = x^2 - 6x + 9.
Step 4: Stretching Factor of 2
To stretch the graph by a factor of 2 parallel to the y-axis, we need to multiply the x-coordinate by 1/2. This can be done by replacing x with (x/2) in the equation.
So, the stretched equation becomes: -y = (x/2)^2 - 6(x/2) + 9.
Simplifying this equation gives us: -y = (1/4)x^2 - 3x + 9.
Finally, we multiply both sides of the equation by -1 to have y as the subject:
y = (-1/4)x^2 + 3x - 9.
Therefore, the equation of the final graph in the form y = ax^2 + bx + c, where a, b, and c are constants, is: y = (-1/4)x^2 + 3x - 9.