Question

Which of the following input-output tables represents the function graphed below?

Answer 1

The graph represents the quadratic function f(x) = 1/2 * x^2 - 2, and the appropriate input-output table should reflect the corresponding x and y values from this function for the given parabola.

The graph illustrates a quadratic function, forming a upward-facing parabola with its vertex at (0, -2) and crossing the x-axis at (2, 0) and (-2, 0). This suggests a quadratic function in the form f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.

In this case, the vertex form of the quadratic function is f(x) = a(x - 0)^2 - 2, simplifying to f(x) = ax^2 - 2.

Using the x-axis point (2, 0), we find 0 = a(2)^2 - 2, leading to a = 1/2.

Therefore, the function graphed is f(x) = 1/2 * x^2 - 2.

Analyzing the provided input-output tables, the one aligning with the x and y values of f(x) corresponds to the graphed function.

Answer 2

The function that represents the graph is
`y = -2 + (1)/(2)\cdot x^(2)`.

Explanation:

The graphic represents a vertical parabola, whose standard equation is:

`y-k = C\cdot (x-h)^(2)`

Where:

`x` - Independent variable, dimensionless.

`y` - Depedent variable, dimensionless.

`h` - Horizontal component of the vertex, dimensionless.

`k` - Vertical component of the vertex, dimensionless.

`C` - Vertex constant, dimensionless. Where
`C > 0` when vertex is an absolute minimum, otherwise it is an absolute maximum.

According to the figure, vertex is located in
`(0, -2)`. Now we determine the vertex constant by using the following values in the standard equation:

`x = -2`,
`y = 0`,
`h = 0`,
`k = -2`

`C = (y-k)/((x-h)^(2))`

`C = (0-(-2))/((-2-0)^(2))`

`C = (2)/(4)`

`C = (1)/(2)`

The function that represents the graph is
`y = -2 + (1)/(2)\cdot x^(2)`.