We are given a table of x and y values and we are asked to find a possible function that represents it.
We notice that the function has a symmetry around the y axis, since the relationship shows:
when x = 1 , y = 2
when x = -1, y = 2
and also
when x = 3 , y = 10
when x = - 3, y = 10
(same value of y for opposite values of x)
In fact, this function could be well represented by a quadratic function of thr form:
f(x) = x^2 + 1
since
f(1) = 1^2 + 1 = 2
f(-1) = (-1)^2 + 1 = 2
f(3) = 3^2 + 1 = 10
f(-3) = (-3)^2 + 1 = 10
The second function that comes expressed as points on the plane, and given by the following 5 points:
(-9,0), (-4, 5), (0, 9), (4, 5), and (9,0)
is again a function that shows a symmetry around the y-axis, so it is an even function again, and loos like the vertex of it is located at the point (0.9)
notice as well, that as we move away from the y-axis in the x values, the values for y decrease (are less and less than 9, as the x value lies further away from the y-axis.
Then we suspect that the function representing this, is an absolute value function with vertex at the point (0, 9) on the y-axis.
Notice that the "slope" of this function is (to the right of the y axis) given by:
slope = (y2 - y1) / (x2 - x1) = (0 - 5) / (9 - 4) = -5 / 5 = -1
and on the left of the y-axis, the slope is "1".
Then, we use the following absolute value function:
f(x) = - |x| + 9
we now chek the given pairs :
f(-9) = - |-9| + 9 = 0 correct!
f(-4) = - |-4| + 9 = - 4 + 9 = 5 correct!
f(0) = - |0| + 9 = 9 correct!
f(4) = - |4| + 9 = -4 + 9 = 5 correct!
f(9) = - |9| + 9 = -9 + 9 = 0 correct!
Then the function we suggest for this second case is:
f(x) = - |x| + 9