Question

Determine which situations best represent the scenario shown in the graph of the quadratic functions, y = x2 and y = x2 + 3. Select all that apply.

The quadratic function, y = x2, has an x-intercept at the origin

The quadratic function, y = x2 + 3, has an x-intercept at the origin

From x = -2 to x = 0, the average rate of change for both functions is positive

From x = -2 to x = 0, the average rate of change for both functions is negative

For the quadratic function, y = x2, the coordinate (2, 3) is a solution to the equation of the function.

For the quadratic function, y = x2 + 3, the coordinate (2, 7) is a solution to the equation of the function.

Answer 1

The quadratic function,
`y=x^(2)`, has an x-intercept at the origin

From
`x = -2` to
`x = 0`, the average rate of change for both functions is negative

For the quadratic function,
`y=x^(2)+3`, the coordinate
`(2, 7)` is a solution to the equation of the function

Explanation:

we have the quadratic functions

`y=x^(2)`

`y=x^(2)+3`

Verify each statement

case A) The quadratic function,
`y=x^(2)`, has an x-intercept at the origin

Remember that

The x-intercept is the value of x when the value of y is equal to zero

The point
`(0,0)` represent and x-intercept and a y-intercept

therefore

The statement is True

case B) The quadratic function,
`y=x^(2)+3`, has an x-intercept at the origin

The quadratic function,
`y=x^(2)+3` has no x-intercept

so

The statement is False

case C) From
`x = -2` to
`x = 0`, the average rate of change for both functions is positive

Observing the graph from
`x = -2` to
`x = 0`, the average rate of change for both functions is negative

so

The statement is False

case D) From
`x = -2` to
`x = 0`, the average rate of change for both functions is negative

The statement is True

case E) For the quadratic function,
`y=x^(2)`, the coordinate
`(2, 3)` is a solution to the equation of the function

we know that

If a ordered pair is a solution of the quadratic function

then

the ordered pair must be satisfy the quadratic equation

Substitute the value of x and the value of y in the quadratic function and then compare

`3=2^(2)`

`3=4` ------> is not true

The ordered pair is not a solution

so

The statement is False

case F) For the quadratic function,
`y=x^(2)+3`, the coordinate
`(2, 7)` is a solution to the equation of the function

we know that

If a ordered pair is a solution of the quadratic function

then

the ordered pair must be satisfy the quadratic equation

Substitute the value of x and the value of y in the quadratic function and then compare

`7=2^(2)+3`

`7=7` ------> is true

The ordered pair is a solution

so

The statement is True

Answer 2

The best represent the scenario shown in the graph of the quadratic functions, y = x2 and y = x2 + 3 are the following:
The quadratic function, y = x2, has an x-intercept at the origin

From x = -2 to x = 0, the average rate of change for both functions is negative

For the quadratic function, y = x2 + 3, the coordinate (2, 7) is a solution to the equation of the function.