SOLUTION
Step 1: Determine the domain
The domain isfx:x 2 Rgbecause there is no value for whichg(x)is undefined.Step 2: Determine the range
The range ofg(x)can be calculated as follows:x^2 0
x^2 + 2 2
g(x) 2Therefore the range isfg(x) :g(x) 2 g.Intercepts
They-intercept:
Every point on they-axis has anx-coordinate of 0, therefore to calculate they-intercept letx= 0.
For example, they-intercept ofg(x) =x^2 + 2is given by settingx= 0:
g(x) =x^2 + 2
g(0) = 0^2 + 2
= 2This gives the point(0; 2).
Thex-intercepts:
Every point on thex-axis has ay-coordinate of 0, therefore to calculate thex-intercept lety= 0.
For example, thex-intercepts ofg(x) =x^2 + 2are given by settingy= 0:
g(x) =x^2 + 2
0 =x^2 + 2
2 =x^2There is no real solution, therefore the graph ofg(x) =x^2 + 2does not havex-intercepts.
Turning points
The turning point of the function of the formf(x) =ax^2 +qis determined by examining the range of the
function.
- Ifa > 0 , the graph off(x)is a “smile” and has a minimum turning point at(0;q).
- Ifa < 0 , the graph off(x)is a “frown” and has a maximum turning point at(0;q).
Axes of symmetry
The axis of symmetry for functions of the formf(x) =ax^2 +qis they-axis, which is the linex= 0.
162 6.3. Quadratic functions