Axis of symmetry, Increasing/decreasing interval , domaine , range etc .

Question

Hi , need need some help with this question. 

 

For y=5x -3 , what is the range, domaine , min/max , equation of axis of symmetry and the increasing/decreasing intervals. 

 

For y= 2( x-3)^2 +1 , what is the range, domaine , min/max , equation of axis of symmetry and the increasing/decreasing intervals. 

Answer 1

The domain is the set of x values in which a function is defined.  The range is the set of y values in which a function is defined.

 

 

y = 5x - 3      is a linear function.  Linear functions have domain and range of all real numbers.  Linear functions have no axis of symmetry.  Since the linear function has a positive slope, the function increases in the interval (-∞, ∞).

 

 

y = 2(x - 3)² + 1      is a quadratic function written in vertex from.  A quadratic function has a domain of all real numbers.  The range is determined by the vertex of the function.  The orientation of the graph that represents the function determines whether there is a minimum or maximum.

 

A quadratic function in vertex form is

 

y = a(x - h)² + k

 

where:

a is the coefficient of the x² term

vertex is the coordinate (h, k)

 

 

From the function,

 

a = 2

h = 3

k = 1

 

 

The vertex is (3, 1).  Since the value of a is positive, the parabola opens upward, making the vertex a minimum point.

 

So the range is all values greater than or equal to 1.

 

Axis of symmetry is  x=3.

 

Decreasing intervals is  (-∞, 2).

Increasing intervals is  (2, ∞).