Section 8.5: Using intercept form Heather Durfee Elizabeth Clifford Joey Lograsso

graphing f(x)=a(x-p)(x-q)

Before we start Make sure you understand what the problem wants you to find. Make sure you recognize if the numbers in the problem are positive or negative. Make sure you understand f(x) or in this case h(x) is the same thing as y. If the numbers were negative the equation would be f(x)= a(x+p)(x+q). These numbers are positive because the intercept form equation is f(x)= a(x-p)(x-q).

Step One: Finding the x intercepts The x- intercepts are also called p and q. p= 7 q= 3 Plot the points on your graph.

Step Two: Find the Axis of Symmetry To figure out the vertex you must figure out the axis of symmetry because the vertex lies on it. The equation to find the axis of symmetry is x=p+q/2. Plug in the parts you know (p(7) and q(3)); x=7+3/2. Solve for x. x=7+3/2 x=10/2 x=5 Plot on graph with a dotted line.

Step Three: Find Vertex We know the vertex lies on the axis of symmetry, so to find the vertex you simply have to plug in 5 for x. h(x)= -4(5-7)(5-3) h(x)= -4(-2)(2) h(x)= -4(-4) h(x)= 16 So the vertex is (5,16)

Step Four: Plotting more points To do this you must make a table of values. Then just plot the points! x 3 4 5 6 7 y 12 16

Step Five: Answering the Questions Now just connect the dots! vertex: (5,16) Axis of Symmetry: x=5 x-intercepts: p=3; q=7 Domain: all real numbers Range: y≤16

Finding zeros of a function

Before we start The zeros are also the x- intercepts. If you get an equation like f(x)=ax²+bx+c you must factor to f(x)=a(x-p)(x-q). f(x) is the same thing a y.

Part One Substitute 0 for f(x). Set each parenthesis equal to 0. 0=1/3(x+5)(x-1) 0=x+5 0=x-1 -5 +1 x=1/3 x=-5 x=1

Part Two: Step One: Factoring Find a greatest common factor and if you can’t then multiply a*c. Then find factors of a*c that add up to b. Group x³ and x² and -9x and 9 together. Factor out x² and -9. Group x² and -9 together Realize that x² and 9 are both squared and factor it. y=(x³-x²)(-9x+9) y=x²(x-1)-9(x-1) y=(x²-9)(x-1) y=(x+3)(x-3)(x-1)

Step Two: Setting each to 0 Now you just set each parenthesis equal to 0. y=(x+3)(x-3)(x-1) o=x+3 0=x-3 0=x-1 -3 +3 +1 x=-3 x=3 x=1

Writing quadratic functions

Step One: Which Equation? Now this is a good question…. There are many possible different equations but in this chapter we have learned about three specific ones; intercept form, standard form, and vertex form. It depends on what your looking for If your working with a vertex then obviously vertex form. If your working with intercepts then intercept form. If your working with points that the parabola passes through it all just depends.

Part One: Vertex Plug in the points that you are given. Foil (x-4)². Combine like terms. f(x)=a(x-h)²+k f(x)=1(x-4)²+8 f(x)=(x-4)(x-4)+8 f(x)=x²-8x+16+8 f(x)=x²-8x+24

Part Two: Passes through In this equation you would you use intercept form. Plug in the points you already know. Plug in the point (-4,3) to figure out a. Solve for a. y=a(x-p)(x-q) y=a(x+5)(x+1) 3=a(-4+5)(-4+1) 3=a(-3) a=-1 y=-(x+5)(x+1)

Real life application