So, if we see the basic equation y=mx+b, we see that m=2, and b=1. If you look closely, this is basic rotation and translation of the original graph. First, I would translate the "mother graph" (y=mx) and then translate one up. Then, we would rotate the graph to the right two units.
The translation is vertical if the added term is outside the main function and horizontal if it is inside it, next to the x. For example, y = x^2 represents a parabola, with its lowest point at (0,0). If we have the equation y = x^2 + 2 then we have translated the parabola up two units -- its lowest point is now x = 0, y = 2. But if we write y = (x + 2)^2, then we are translating two units to the left, and the lowest point is x = -2, y = 0.
To translate the graph of ( y = -x^2 ) to produce the graph of ( y = -(x-2)^2 ), you would shift the graph 2 units to the right. This transformation occurs because the expression inside the parentheses, ( (x-2) ), indicates a horizontal shift. The negative sign in front of the squared term indicates that the parabola opens downward, which remains unchanged in the translation. Thus, the vertex moves from the origin (0, 0) to the new vertex at (2, 0).
The graph would be translated upwards by 2 units.
To shift a graph of a function ( f(x) ) upward by ( k ) units, you simply add ( k ) to the function. The new function becomes ( f(x) + k ). For example, if the original function is ( f(x) = x^2 ) and you want to shift it up by 3 units, the new function would be ( f(x) + 3 = x^2 + 3 ). This transformation moves every point on the graph up by the specified amount.
Y=|x+2|
the translation of 2 is the one that triangle moves by 4 units right and 8 units up
If you we're at the point (8,-2) and you went 5 units right and 2 units up, you'd be at (13,0).
(x,y) > (x + 8, y + 2)
So, if we see the basic equation y=mx+b, we see that m=2, and b=1. If you look closely, this is basic rotation and translation of the original graph. First, I would translate the "mother graph" (y=mx) and then translate one up. Then, we would rotate the graph to the right two units.
f(x) cannnot be a graph of itself translated down by anything other than 0 units.
The translation is vertical if the added term is outside the main function and horizontal if it is inside it, next to the x. For example, y = x^2 represents a parabola, with its lowest point at (0,0). If we have the equation y = x^2 + 2 then we have translated the parabola up two units -- its lowest point is now x = 0, y = 2. But if we write y = (x + 2)^2, then we are translating two units to the left, and the lowest point is x = -2, y = 0.
To translate the graph of ( y = -x^2 ) to produce the graph of ( y = -(x-2)^2 ), you would shift the graph 2 units to the right. This transformation occurs because the expression inside the parentheses, ( (x-2) ), indicates a horizontal shift. The negative sign in front of the squared term indicates that the parabola opens downward, which remains unchanged in the translation. Thus, the vertex moves from the origin (0, 0) to the new vertex at (2, 0).
The graph of is shifted 3 units down and 2 units right. Which equation represents the new graph?
The graph would be translated upwards by 2 units.
(2,1)
To shift a graph of a function ( f(x) ) upward by ( k ) units, you simply add ( k ) to the function. The new function becomes ( f(x) + k ). For example, if the original function is ( f(x) = x^2 ) and you want to shift it up by 3 units, the new function would be ( f(x) + 3 = x^2 + 3 ). This transformation moves every point on the graph up by the specified amount.