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Question

function y= -x^2 - 3x^2 - 1, determine the following characteristic gor this function.

 

- number of x intercepts

 

-number of y intercepts

 

- number of turing point

 

- range and domain

 

- End behavior

 

 

 

Answer 1

The quadratic function y = - x²  - 3x²  - 1

y = - 4x²  - 1

Compare it to y = ax²  + bx  + c

a = - 4, b = 0, c = - 1

b² - 4ac = (0)² - 4(- 4)( - 1)

= - 16

Discriminant b² - 4ac is negative means the equation has two imaginary roots.

A quadratic function represents a parabola in the graph.

There is no x intercepts.

Answer 2

The quadratic function y = - x²  - 3x²  - 1

To find y intercept substitute x = 0 in y = - x²  - 3x²  - 1.

y = - (0)²  - 3(0)²  - 1

y = - 1

y intercept of the function is - 1.

Number of y intercepts = 1.

Answer 3

End behavior

The polynomial function y = - x²  - 3x²  - 1

degree of the function is 2 and leading coefficient is - 1.

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

All even degree polynomials are either up on both ends and or down on both ends.depending on whether the polynomial has, respectively, a positive or negative leading coefficient.

The above polynomial even degree  polynomial with a negative leading coefficient .

So the graph down on both ends.

Answer 4

The quadratic function  y = - x²  - 3x²  - 1

y = - 4x² - 1

Compare it to y = ax²  + bx  + c

a = - 4, b = 0, c = - 1

The quadratic function has only one turning point at vertex.

In this case a is negative means the parabola opens down.

When the parabola opens down and it has a maximum point( turning) which is the vertex of parabola.

To find x coordinate of vertex x = -b/2a

x = - 0/2(- 4)

x = 0

To find y coordinate of vertex substitute x = 0 in y = - 4x²  - 1.

y = - 4(0)²  - 1

y = - 1

Maximum  point at (x, y) = (0  -1)

Turning point is (0 , - 1).

Answer 5

The quadratic function  y = - x²  - 3x²  - 1

y = - 4x² - 1

Compare it to y = ax²  + bx  + c

a = - 4, b = 0, c = - 1

In this case a is negative means the parabola opens down.

When the parabola opens down and it has a minimum point which is the vertex of parabola.

To find x coordinate of vertex x = -b/2a

x = - 0/2(- 4)

x = 0

To find y coordinate of vertex substitute x = 0 in y = - 4x²  - 1.

y = - 4(0)²  - 1

y = - 1

When the parabola opens down it has a maximum point which is the vertex of parabola (0, - 1)

We know that domain of the function is all possible x values and range is all possible y values.

Parabola domain x =  all real numbers.

In the maximum point y = - 1  so the graph of parabola cannot be upper than - 1.

Thus the range of function y ≤ - 1.

Domain of function is all real numbers.

Range of the function is  {y |y ≤ - 1}.