Quadratics Final
- 1. QUADRATICS Three Forms and Their Graphs
- 2. Standard Form ax²+bx+c Special Factoring Patterns • Difference of Two Squares: a²-b²=(a+b)(a-b) example: x²-16=(x+4)(x-4) The square root of x² is x and the square root of 16 is 4. • Perfect Square Trinomials: a²+2ab+b²=(a+b)², a²- 2ab+b²=(a-b)² example: x²+14x+49=(x+7)² The square root of 49 is 7, and 7x2 is 14. example: x² -6x+9=(x-3)² The square root of 9 is 3, and -3x2 is -6.
- 3. Graphing in Standard Form Characteristics of the graph: • The graph opens up if a>0, and opens down if a<0 • The graph is narrower than the graph of y=x² if lal>1 and wider if lal<1 • The axis of symmetry is x=-b/2a, and the vertex has x-coordinate –b/2a • The y-intercept is c, so the point (0,c) is on the parabola Steps For Graphing: 1. Identify the coefficients a,b, and c. 2. Find the vertex: x=-b/2a, then calculate the y-coordinate by plugging the value of x back into the function. Plot the vertex (-b/2a, f(-b/2a)). 3. Draw the axis of symmetry x=-b/2a 4. Identify the y-intercept c, and plot the point (0,c). Reflect this point in the axis of symmetry to plot another point. 5. Evaluate the function for another value of x, such as x=1. Plug 1 into the function and solve. Plot the point and its reflection in the axis of symmetry. 6. Draw a parabola through the plotted points
- 4. Graphing in Vertex Form y=a(x-h)²+k Characteristics of graph: • a tells whether the graph opens up or down, and whether it is wide or narrow • h tells how far left or right the parabola is • k tells how far up or down the parabola is • vertex is (h,k) Steps for Graphing: 1. Determine the vertex (h,k). 2. Plot the vertex and draw the axis of symmetry. ~the equation of the axis of symmetry is x=h. 3. Find the coordinates to the left and right of the vertex. ~go over left or right by one unit, and then up or down by the value of a. This trick only works once. 4. Connect the points and label them.
- 5. Graphing in Intercept Form y=a(x-p)(x-q) Characteristics of Graph: • a tells you whether the graph opens up or down, and whether it is wide or narrow • The x-intercepts are p and q • The y-intercept is apq Steps for Graphing: 1. Plot the x-intercepts 2. Find the coordinates of the vertex (p+q/2, f(p+q/2)) 3. Plot the vertex and axis of symmetry ~the equation for the axis of symmetry is x=p+q/2 4. Connect the points and label
- 6. Converting to Standard Form • Vertex to Standard: multiply out and collect like terms y=-3(x-2)²+2 → y=ax²+bx+c (x-2)(x-2)=x²-2x-2x+4=x²+4x+4 y=-3(x²-4x+4)+2 y=-3x²+12x-12+2 y=-3x²+12x-10 • Intercept to Standard: multiply out and collect like terms y=3(x-2)(x-5) → y=ax²+bx+c y=3(x²-7x+10) y=3x²-21x+30