Algebra and Trigonometry question. Serious replies only! Please show work. Thank you so much!!?

Question

The graph of f (x) = 2 (x-1) ^2 +3 has the following characteristics.

a). vertex: (1,3) and opens downward
b). vertex: (1,3) and doesn't cross the x axis
c).vertex: (1,3) and has two y intercepts
d). vertex: (1,3) and y intercept less than 5
e). none of these

Answer 1

Answer b is the correct choice.

This quadratic function is in vertex form,
f(x) = a (x - h) + k, where (h,k) is the vertex.

Thus, the vertex of your function,
f(x) = 2 (x-1) ^2 +3, is the point (1,3).

When the leading coefficient, a, is positive, the graph of the function (a parabola) opens upward like an uppercase letter U. When a is negative, it opens downward (flip the U upside down).

Your a is positive (2) so it opens up and since it's vertex (the bottom of the parabola in this case) is 1,3 and already above the x-axis, it will never cross the x-axis.

All the other choices are incorrect. C will never be true for any quadratic function (but is true for square root equations) and in fact is not true for any function at all because it would not pass the vertical line test.

By the way, this is purely an algebra question and doesn't involve any trigonometry.

Answer 2

f' = 4(x-1) =0
4x = 4
x=1
y =f(1) = 2(1-1)^2 +3 = 3

this shows the vertex is 1,3 and since the first derivative is positive the parabola opens upward

f(x) = 0 =2(x-1)^2 +3
-3 = 2(x^2 -2x +1)
-3 = 2x^2 -4x +2
0 = 2x^2 -4x +5
x= (4 +/-sqrt(16-40))/4 since the roots are obviously imaginary when f(x) =0 , that is, at the x axis the curve does not cross the x axis

the answer is b.)

Answer 3

the 2(...) is positive, so graph opens upward, and ..(..-1) + 3 makes the vertex (1,3), so it can't cross x-axis.

answer b

Answer 4

I won't do the question for you but if you:

1. Isolate the f by dividing both sides by x giving you f=(2(x-1)^2+3)/x

2.repace the x with an integer such as 0,1,2,3,-1,-2,-3

3.work out the equation for it 3 or 4 times with these numbers you should be able to plot your graph and get your answer

Answer 5

b)

Graph it.

Another way to know that it doesn't cross the x-axis is to set the equation = 0 and solve for x.

2x² - 4x + 5 = 0

x = 1 +/- sqrt(-6)/2 { Quadratic Formula }

Since there is no real value for sqrt(-6), the graph never crosses the x-axis.