Question

Without using a graphing calculator, sketch the graphs of the functions below based on transformations of the

Answer 1

Answer:

See explanation and figures below.

Step-by-step explanation:

a. h(x) = (x + 2)^2 − 4

For this case the original function is $y = x^2$ with a vertex (0,0) for the new function we have the following general expression $h(x)= (x-h)^2 + k$ on this case we see that h = -2 and k =-4  so then the vertex for the new function would be (-2,4), and the x intercepts are (-4,0) and (0,0) as we can see on the figure attached 1.

b. ℎ(x) = −(x − 4)^2 + 2

For this case the original function is $y = x^2$ with a vertex (0,0) for the new function we have the following general expression $h(x)= (x-h)^2 + k$ on this case we see that h = 4 and k =2  so then the vertex for the new function would be (4,2), and the x intercepts are (2.586,0) and (5.414,0) as we can see on the figure attached 2.

On this case since we have a negative on the quadratic term we see that the parabola would open downwards.  

c. h(x) = 2x^2 − 12x + 19

For this case after the first transformation $2x^2$ we have a compression on the x axis, the -12x would cause that the function move to the right and upward and an expansion on the x axis. And finally the +19 cause that the function moves upward 19 units. The new vertex would be (3,1) and we just have one y intercept (0,19) and none x intercepts. We see the plot on figure 3.

d. p(x) = −2x^2 − 4x − 5

For this case after the first transformation $-2x^2$ we have a compression on the x axis and the parabola would open downwards, the -4x would cause that the function move to the left and down and an expansion on the x axis. And finally the -5 cause that the function moves downward 5 units. The new vertex would be (-1,3) and we just have one y intercept  (0,-5) and none x intercepts. We see the plot on figure 4.

e. h(x) = 3x^2 + 6x

For this case after the first transformation $3x^2$ we have a compression on the x axis, the +6x would cause that the function move to the left and down and an expansion on the x axis. The new vertex would be (-1,-3) and the x intercepts would be:

$3x(x+2) = 0$

$x=0 , x=-2$

And we can see the plot on figure 5 attached.