Question

comparing properties of quadratic functions
function 1

x	y
4	-8
1	1
-2	4
-5	1
-8	-8
-11	-23

function 2
graph of a parabola
(a) what is the vertex of function 1? ( , )
(b) what is the vertex of function 2? ( , )
(c) which function has the larger maximum value?
function 1
function 2
what is the larger maximum value?

Explanation:

Step1: Find vertex of Function 1

For a quadratic function (symmetric), find the axis of symmetry. The x - values \(x = 1\) and \(x=-5\) have the same \(y = 1\), \(x = 4\) and \(x=-8\) have the same \(y=-8\). The midpoint of \(x = 1\) and \(x=-5\) is \(\frac{1+( - 5)}{2}=\frac{-4}{2}=-2\). At \(x=-2\), \(y = 4\). So vertex of Function 1 is \((-2,4)\).

Step2: Find vertex of Function 2

The graph of Function 2 is a parabola. The vertex is the highest point. From the graph, the vertex is at \((3,5)\) (by looking at the peak of the parabola).

Step3: Compare maximum values

Function 1's maximum (vertex y - value) is \(4\). Function 2's maximum (vertex y - value) is \(5\). So Function 2 has a larger maximum value of \(5\).

Answer:

(a) \((-2, 4)\)
(b) \((3, 5)\)
(c) Function 2; The larger maximum value is \(5\)