4. Given $y=3 x^{2}-24 x+36$ :
(a) (5 points) Find the $x$ and $y$ intercepts.(Do not round)
(b) (4 points) Find the vertex. (Do not round)
(c) (5 points) Use the information from part a and b to graph label your points.
Answer
The final answer is: The x-intercepts of the function are \(x=2\) and \(x=6\), the y-intercept is \(y=36\), and the vertex of the function is at the point \((4, -12)\). So, the final answer is \(\boxed{(2, 6, 36, (4, -12))}\).
Step 1 ：First, we need to find the x and y intercepts of the function \(y=3 x^{2}-24 x+36\). The x-intercepts of a function are the values of x where y is equal to 0. To find these, we can set the function equal to 0 and solve for x. The y-intercept of a function is the value of y when x is equal to 0. To find this, we can substitute 0 for x in the function and solve for y.
Step 2 ：The x-intercepts of the function are 2 and 6, and the y-intercept is 36.
Step 3 ：Next, we need to find the vertex of the function. The vertex of a parabola given by the function \(y=a(x-h)^{2}+k\) is the point (h, k). In this case, the function is in the form \(y=ax^{2}+bx+c\), so we can use the formula \(h=-\frac{b}{2a}\) to find the x-coordinate of the vertex, and then substitute this value into the function to find the y-coordinate.
Step 4 ：The vertex of the function is at the point (4, -12).
Step 5 ：Finally, we can use this information to graph the function. We can plot the x and y intercepts and the vertex, and then sketch the curve of the function.
Step 6 ：The final answer is: The x-intercepts of the function are \(x=2\) and \(x=6\), the y-intercept is \(y=36\), and the vertex of the function is at the point \((4, -12)\). So, the final answer is \(\boxed{(2, 6, 36, (4, -12))}\).
