helppp please!4

Question

function y=x^2 - 4x - 5, 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

(1)

y = x² - 4x - 5 .

To determine the number of x-intercepts of the graph of a quadratic function, substitute 0 for y  .

0 = x² - 4x - 5

 x² - 4x - 5 = 0

 x² - 5x + x - 5 = 0

x(x-5) +1(x-5) = 0

(x-5)(x+1) = 0

x-5 = 0 and x+1 = 0

x = 5 and x = -1

So the function has two x-intercepts .

Answer 2

(2)

y = x² - 4x - 5 .

To determine the number of y-intercepts of the graph of a quadratic function, substitute 0 for x  .

y = 0² - 4(0) - 5 

y = 0 - 0 -5

y =-5

So the function has one y-intercepts .

Answer 3

 

 (3)

The function is y = x² - 4x - 5 .

Turning points are nothing but the local minimum / maximum .

To find the local minimum / maximum , equate the first derivative to zero .

y' = 2x - 4

 2x - 4 = 0

2x = 4

x = 4/2

x = 2

So the function has one turning point at x = 2 .

 

Answer 4

 

 (4)

The function is y = x² - 4x - 5 .

We first put the equation in to the form for a translated parabola y = a (x - h )^2 + k .

Rewrite the function 

y = x² - 4x - 5

y = x² - 4x + 4 - 4 - 5

y = x² - 4x + 4 - 9

y = (x-2)² - 9

The above function represents a parabola vertex form  y = a (x - h )^2 + k .

 a  = 1 , h  = 2 and k  = -9 .

a  is positive number the parabola opens up and has minimum value.

When the parabola opens up it has a minimum point which is the vertex of parabola ( 2, -9)

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

 parabola domain x  =  all real numbers.

In the minimum point y  = -9  so the graph of parabola cannot be lower than -9.

Thus the range of function y  ≥ -9.

Domain of function is all real numbers.

Range of the function is  {y |y  ≥ -9}. 

 

Answer 5

(5)

The function is y = x² - 4x - 5 .

The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

The given function has the even degree and leading coefficient is positive .

 So, the end behavior is: