1. Functions and Their Graphs
6.10
Functions and Their Graphs

2. Relation and Function A relation is any set of ordered pairs.
A function is a special type of relation where each value of the independent variable corresponds to a unique value of the dependent variable.
Domain the set of values used for the independent variable.
Range The resulting set of values obtained for the dependent variable.

3. Vertical Line Test
If a vertical line can be drawn so that it intersects the graph at more than one point, then each x does not have a unique y.

4. Practice P. 374 # 18 & 20

5. Types of Functions
Linear: y = ax + b
Quadratic: y = ax2 + bx + c

6. Graphs of Quadratic Functions
axis of symmetry
vertex
vertex

7. Graphs of Quadratic Functions continued
Axis of Symmetry of a Parabola
This formula also gives the x-coordinate of the vertex of a parabola. Substituting this value into the given quadratic equation and evaluating the equation yields the y-coordinate of the vertex.

8. General Procedure to Sketch the Graph of a Quadratic Equation
Determine whether the parabola opens upward or downward.
Determine the equation of the axis of symmetry.
Determine the vertex of the parabola.
Determine the y-intercept by substituting x = 0 into the equation.
Determine the x-intercepts (if they exist) by substituting y = 0 into the equation and solving for x.
Draw the graph, making use of the information gained in steps 1 through 5. Remember the parabola will be symmetric with respect to the axis of symmetry.

9. Example: Graph y = x2 + 2x 3. Since a = 1, the parabola opens up.
Axis:
y-coordinate of vertex
y-intercept:
(-1, -4)
(0, -3)

10. Graph y = x2 + 2x 3 continued
x-intercepts:
Plot the points and sketch.

11. P. 375 # 58

12. Graphs of Exponential Functions
Graph y = 3x.
Domain: all real numbers
Range is y > 0
(3, 1/27)
1/27 3
(2, 1/9)
1/9 2
(1, 1/3)
1/3 1
(3, 27) 27 3 9 1
y = 3x
(2, 9) 2
(1, 3)
(0, 1)
(x, y) x

13. Graphs of Exponential Functions
Domain: all real numbers
Range is y > 0
(3, 1/27)
1/27 3
(2, 1/9)
1/9 2
(1, 1/3)
1/3 1
(3, 27) 27 3 9
(2, 9) 2
(1, 3) 1
(0, 1)
(x, y) x

14. Try P. 375 #74

15. Homework: p. 374 # 11 – 79 eoo
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