Quadratic functions q and w are graphed on the same coordinate grid the veex of the graph of q is 18 units below the vertex of the graph of w. which pair of functions could have been used to create the graphs of q and w?
To construct the two quadratic functions, let's start with the general form of a quadratic function:
f(x) = ax^2 + bx + c
Since the vertex of q is 18 units below the vertex of w, we can deduce that the c-values for q and w are different. Let's denote the c-value for q as c1 and the c-value for w as c2.
Therefore, we have:
q(x) = ax^2 + bx + c1
w(x) = ax^2 + bx + c2
Both quadratic functions have the same a and b values, so only their c-values differ.
To determine the pair of functions, we need to consider how the vertex of a quadratic function relates to its c-value. The x-coordinate of the vertex is given by:
x = -b/2a
Given that the vertex of q is 18 units below the vertex of w, we can express the relationship using the x-coordinate of the vertex. Let's denote the x-coordinate of the vertex of w as h.
Therefore, x-coordinate of the vertex of q = h + 18
Plugging this into the equation for the x-coordinate of the vertex, we have:
(h + 18) = -b/2a
Simplifying, we get:
2ah + 36a = -b
From this equation, we can see that for the two quadratic functions to have the same a and b values but different c-values, h must be non-zero (since we can't divide by zero). This means that h cannot be equal to zero; it must be some non-zero value.
In conclusion, any pair of quadratic functions q(x) = ax^2 + bx + c1 and w(x) = ax^2 + bx + c2, where a and b are the same, but c1 and c2 are different, and h (the x-coordinate of the vertex of w) is non-zero could have been used to create the graphs of q and w.