Math-A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis? f ( x) x x x x x x 3 3 ( x) x We call functions that are symmetric about the y -axis, even functions. Quadratic Function: even f
Analyzing the graph 1. Where is the function increasing?. Where is the function decreasing? 3. Is the function even? What is the axis of symmetry? 4. Are there any extrema? If so, what type are they? 5. How does it relate to its parent function? 6. What is the end behavior of the graph? as x, y? as x, y? 7. What is the domain of the graph? 8. What is the range of the graph? 9. What is the average rate of change between two given values of x?
x x y x y ( x y 3x y 3( x Which of the following are not even? 3) 5) y x 1 Yes 6 4 No, axis of symmetry is x = 3 Yes No, axis of symmetry is x = -5 Yes g( x) x 1 No, axis of symmetry is x = 1 g( x) x Yes 5 x 3 No, axis of symmetry is x =
Average Rate of Change ( x ) What is the average rate of change between x = and 4? (They don t give you the y-values but you can get these from the graph. Means what is the slope of the graph between the two points? (, y 1 ) and (4, y) slope y 4 m x Another way of saying this is: what is the average rate of change on the interval x = (, 4)?
The function is increasing. ( x ) Where is the function increasing? means, (as you go from left to right on the x-axis) on what interval(s) of the x-axis does a tangent line have a positive slope. The slope of a tangent line at any point on the graph for the interval x = (, ) is positive. We say: the function is increasing on the (x) interval: (, ) f ( x) on x (, )
The function is decreasing. ( x ) Where is the function decreasing? means, on what interval(s) of the x-axis does a tangent line have a negative slope? We say: The function is decreasing on the interval (, ) f x on: x = (, )
Graphical Extrema (extreme function values) ( x ) Are there any extrema? means, are there any places on the graph where the tangent line has a slope of zero OR peaks or valleys in the graph Yes, at (, 0) the function value f(x), is an absolute minimum is the minimum y-value
Infinity Notation ( x ) What is the end behavior of the graph? As x gets smaller, y gets bigger. as x, y Up on right, up on left. As x gets bigger, y gets bigger. as x, y As x gets smaller, y gets bigger.
Analyzing Functions Graphically ( x ) What is the domain of the graph? We say the domain is all real numbers. x = (, ) What is the range of the graph? y 0 y [ 0, )
What does the solution to a single variable inequality mean? All values of x that make the inequality true. How do we show the solution on a number line? We shade the numbers that are the solution. A single variable inequality is made up of: (a) boundary points on the number line (b) shading on one side (or the other side) of the boundary. x + 1 7 or x 1 > 7 4 5 6 7 8 9 boundary points: the solution to the equation x = 6 or x = 8 x + 1 = 7 or x 1 = 7 Shading: the solution
A two-variable inequality is made up of (a) a boundary (line, parabola, etc.) on the x-y plane (b) Shading one side of the boundary. y x 3 boundary points: the graph of the equation y = x + 3 Shading: the solution The solution to the inequality is all of the points in one of the half planes. Do the points on the boundary line make the inequality true? Look at the inequality symbol: or : YES < or >: NO
Graph the following inequality. x 3y 6 y ( x ) 3
Solve 0 x 4x 5 0 x 4x 5 Find the boundary numbers Solve the equation: 0 = (x 5)(x + 1) x = 5, -1-1 5 The numbers -1 and 5 divide the solution from the non-solution. 0-1 5 Test "0" 0 (0) 4(0) 0 5 5 0 False shade outside x (, 1) (5, )
System of two linear equations: Two equations (of lines) that each have the same two variables. (in this case x and y ) 3x + y = 7 5x - y = -3 What does the solution of a system of two equations in two variables mean? An ordered pair that makes both equations true. Categories of Solutions: Lines Cross one solution Parallel Lines no solutions Same line every point on the line is a solution.
Three methods of solving: y x 3 1) Solve by graphing. ) Substitution y = y y 3x 1 x + 3 = 3x 1 x x 3 = x 1 +1 + 1 ( 1)(y) = ( 1)(3x 1) 4 = x y = 3x + 1) y = x + 3 = x 0 = x + 4 y Solution: (, 5) x 3 y = + 3 3) Elimination add multiples of the equations to eliminate one of the variables. x = 4 x =
System of Inequalities: More than one -variable inequality graphed on the same x-y plot. y > x - y < -x + Two lines that cross divide the plane into 4 regions. Which region contains the points that are the solution to the system of inequalities? y > x AND y < -x + Solution: the points in the overlap region.