1. Graph y=-2x^2+7 on your paper. Find the domain and range. 2. Graph and find the domain and range. y=x^2-4x+3

Question

please help

Answer 1

I will do the first one for you.  Follow the steps.  Then apply these steps to answer question 2 on your own.

 

 

When you look at this function, you know that you will have a parabola because this is a quadratic function.  When graphing a parabola, you need several points:

 

x-intercepts, y-intercept, and your vertex.

 

Another thing you should know is the domain of a quadratic function is all real numbers.  The range is the set of y values that is determined by its vertex.

 

To find the y-intercept, set x=0 and solve for y.

 

y = 7

 

 

y-intercept is (0, 7).

 

 

To find the x-intercepts, set y=0 and solve for x.

 

0 = -2x² + 7

 

-7 = -2x²

 

3.5 = x²

 

x = -1.87           and            x = 1.87

 

 

Your x-intercepts are (-1.87, 0) and (1.87, 0).

 

 

To find your vertex, we use the formula

 

x = -b/(2a)     to find the x-coordinate of the vertex.

 

where:

a = -2

b = 0

c = 7

 

Since b is zero, the x-coordinate of the vertex is x=0.  You can also find the x-coordinate of the vertex by taking the average of the x-intercepts.  In fact,  the formula    -b/(2a)   is the average of quadratic formula's conjugates when finding the x-intercepts:

 

x₁ = (-b - √(b² - 4ac)) / 2a               and                  x₂ = (-b + √(b² - 4ac)) / 2a   

 

This is because the vertex is an axis of symmetry of the parabola. 

 

Since the x-coordinate of the vertex is zero, the vertex is the same as the y-intercept.

 

So lets review,

 

x-intercepts are  (-1.87, 0) and (1.87, 0)

y-intercept is (0, 7)

vertex is (0, 7)

 

 

Plot these points on the graph.  Then connect them in the path they follow.