Let y=3(x-4)^2 -3

a) Show that y=3x^2 -24x +45

b) For the graph of y find both x intercepts by solving the equation 3x^2 -24x +45 =0

A) direction of opening
B) x intercepts
C) y intercept
D) Axis of Symmetry
E) Vertex
F) max or min

take a look at

http://www.wolframalpha.com/input/?i=3%28x-4%29^2+-3+

a) To show that y = 3x^2 - 24x + 45, we need to expand the expression (x-4)^2 and simplify the equation.

Starting with y = 3(x-4)^2 - 3:
Expanding the square:
y = 3(x^2 - 8x + 16) - 3
Distributing the 3 to each term inside the parentheses:
y = 3x^2 - 24x + 48 - 3
Simplifying:
y = 3x^2 - 24x + 45

Therefore, we have shown that y = 3x^2 - 24x + 45.

b) For the graph of y = 3x^2 - 24x + 45, we can find the x-intercepts by solving the equation 3x^2 - 24x + 45 = 0.

To find the x-intercepts, we need to solve the quadratic equation. We can either factor the equation or use the quadratic formula.

Using factoring:
3x^2 - 24x + 45 = 0
Dividing the equation by 3 to simplify:
x^2 - 8x + 15 = 0
Factoring the quadratic equation:
(x - 3)(x - 5) = 0
Setting each factor equal to zero and solving for x:
x - 3 = 0 or x - 5 = 0
x = 3 or x = 5

A) Direction of opening:
Since the coefficient of x^2 is positive (3 > 0), the parabola opens upwards.

B) X-intercepts:
The x-intercepts correspond to the values of x where y = 0. From our factored equation, we found that x = 3 and x = 5 are the x-intercepts.

C) Y-intercept:
The y-intercept corresponds to the value of y when x = 0. Substituting x = 0 into the equation y = 3x^2 - 24x + 45:
y = 3(0)^2 - 24(0) + 45 = 45
So, the y-intercept is 45.

D) Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex of the parabola. The formula for the axis of symmetry is x = -b/(2a). For our equation, a = 3 and b = -24.
x = -(-24)/(2*3) = 4
So, the axis of symmetry is x = 4.

E) Vertex:
To find the vertex, we can use the formula x = -b/(2a) to find the x-coordinate of the vertex, and then substitute this value into the equation to find the y-coordinate.
x = -(-24)/(2*3) = 4
Substituting x = 4 into the equation y = 3x^2 - 24x + 45:
y = 3(4)^2 - 24(4) + 45
y = 12 - 96 + 45
y = -39
So, the vertex is (4, -39).

F) Max or Min:
Since the parabola opens upwards, the vertex is the minimum point on the graph. Therefore, the graph has a minimum value at the vertex.