For f(x)=ax²+bx+c,

1. if a>0, then the parabola opens upward,
2. if a<0, then the parabola opens downward,
3. the vertex of f is
4. the y-intercept is (0, f(0)) or (0,c),
5. to find the x-intercept(s), we set y=0 and solve for x, so it amounts to solve a quadratic equation ax²+bx+c=0, which suggests that we have three possibilities:
   1. If D=b²-4ac>0, then y=f(x) has two x-intercepts.
   2. If D=b²-4ac=0, then y=f(x) has one x-intercept, which is the vertex.
   3. If D=b²-4ac<0, thne y=f(x) has no x-intercept.

Exercises:

1. For f(x)=3x²-6x+4,
   1. find the vertex of y=f(x),
   2. find the x and y intercepts,
   3. sketch the graph of f.
2. For f(x)=x²-6x+4,
   1. find the vertex of y=f(x),
   2. find the x and y intercepts,
   3. sketch the graph of f.
3. For f(x)=ax²+2x+4, then
   1. find a so that the graph of y=f(x) has exactly one x-intercept,
   2. find a so that the graph of y=f(x) has two x-intercepts,
   3. find a so that the graph of y=f(x) has no x-intercept.
4. For f(x)=-x²+2x+c, then
   1. find c so that the graph of y=f(x) has exactly one x-intercept,
   2. find c so that the graph of y=f(x) has two x-intercepts,
   3. find c so that the graph of y=f(x) has no x-intercept.

Graphs and Inequalities for quadratic functions.

1. For
   1. find the vertex of the graph of y=f(x), (hint: f(x)=x²-6x+8)
   2. find the x and y-intercepts for the graphs of y=f(x),
   3. sketch the graph of f, (Maple syntax: plot((x-2)*(x-4),x=0..6);)
   4. find the interval(s) where f(x)>0 or f(x)<0. [This is to solve x so that or By solving the inequailities, we get f(x)>0 when x and f(x)<0 when
2. For
   1. find the vertex of the graph of y=f(x), (hint: f(x)=-x²+6x-8)
   2. find the x and y-intercepts for the graphs of y=f(x),
   3. sketch the graph of f, (Maple syntax: plot(-(x-2)*(x-4),x=0..6);)
   4. find the interval(s) where f(x)>0 or f(x)<0. [This is to solve x so that or By solving the inequailities, we get f(x)>0 when x and f(x)<0 when

 
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