The function f(x)=x2+6x+18 can be obtained from an even function g by shifting its graph horizontally and vertically. That even function is g(x)= ?

where are the zeros of this parabola?

x^2 + 6 x + 18 = 0

x = [-6 +/- sqrt (36-72)]/2
= -3 +/- 3

vertex is at x = -3

g has it at at x = 0

how high is it? roots are not real so above x axis

y = (-3)^2 -18 +18
= 9

so to be even
g(x) = x^2 + k
when x = 0, y must be 9
so
g(x) = x^2 + 9
=====================
move that left 3
x' = x+3
x^2 + 6 x + 9
move that up 9
x^2 + 6 x + 18

g(x) = x^2

extra credit: shifted by how much in each direction?

left 3, up 9... but the g(x) function is wrong or i might have a glitch in my online assignment

well g = x^2 + anything will work. Steve used zero

totally lost

To obtain the function f(x) = x^2 + 6x + 18 from an even function g(x) by shifting its graph horizontally and vertically, we need to understand the properties of even functions and how shifting affects them.

An even function is symmetric with respect to the y-axis. This means that if we reflect any point (x, y) on the graph across the y-axis, we will get the same point (-x, y). Mathematically, it can be defined as: g(x) = g(-x) for all x in the domain.

To shift the graph of an even function horizontally, we can add or subtract a value from the input (x) of the function. Let's consider a horizontal shift of h units. In this case, g(x) will become g(x - h). This shift will cause all the x-values on the graph to move h units to the right (for positive h) or the left (for negative h).

To shift the graph of an even function vertically, we can add or subtract a value from the output (y) of the function. Let's consider a vertical shift of k units. In this case, g(x) will become g(x) + k. This shift will cause all the y-values on the graph to move k units up (for positive k) or down (for negative k).

For the function f(x) = x^2 + 6x + 18, we can see that it is a quadratic function. By comparing it with the general form of a quadratic function g(x) = ax^2 + bx + c, we can extract the values of a, b, and c.

In this case, a = 1, b = 6, and c = 18. Since f(x) is not an even function, we need to transform it into an even function by shifting its graph horizontally and vertically.

To shift the graph of f(x) horizontally, we need to take the opposite sign of the coefficient of x (b) and divide it by 2. This will give us the value of h for the horizontal shift. In this case, h = -6/2 = -3.

To shift the graph of f(x) vertically, we need to find the constant term (c). The vertical shift value (k) will be -c. In this case, k = -18.

Therefore, the even function g(x) that can be obtained from f(x) = x^2 + 6x + 18 by shifting its graph horizontally by -3 units and vertically by -18 units is:

g(x) = (x + 3)^2 - 18