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
