Quadratic Functions

A quadratic function is a 2nd-degree polynomial function:

$f(x)=ax^2+bx+c$, where $a\ne0$.

The graph of a quadratic function is a parabola, which opens up if $a>0$ and opens down if $a<0$.

The x-coordinate of the vertex of the parabola is given by $x=-\frac{b}{2a}$, and the y-coordinate can be found by substituting this value for $x$ into $f(x)$.

If the vertex of the parabola has coordinates $(h,k)$, then the standard equation of the parabola has the form $y-k=a(x-h)^2$.

The x-intercepts of the parabola, if there are any, are the solutions of the quadratic equation $ax^2+bx+c=0$.

Ex 1 Find the vertex of the parabola $y=2x^2-8x+7$.

Sol We have that $x=-\frac{b}{2a}=-(-8)/(2*2)=2$, and then $y=2(2^2)-8(2)+7=8-16+7=-1$.

Ex 2 Find the minimum value of the function $f(x)=2x^2+2x+5$.

Sol The minimum value of this function is given by the y-coordinate of the vertex. Since the x-coordinate of the vertex is given by $x=-\frac{b}{2a}=-2/(2*2)=-1/2$, the minimum value is given by $y=2(-1/2)^2+2(-1/2)+5=2(1/4)-1+5=1/2+4=9/2$.

Pr 1 Find the vertex of the parabola $y=x^2+6x-2$.

Pr 2 Find the maximum value of the function $f(x)=-x^2-5x+9$.

Pr 3 Find a quadratic function $f(x)$ which has 5 and 1 as the x-intercepts of its graph and which has a minimum value of -12.

Pr 4 Find a parabola $y=ax^2+bx+c$ which has its vertex at the point $(-2,3)$ and which passes through the point $(2,-1)$.

Pr 5 Find the minimum value of the function $g(x)=x^4-8x^2+25$, and find the values of $x$ for which $g(x)$ is a minimum.

Pr 6 Find the vertex of the parabola $x=y^2-10y+40$.

Pr 7 Find a quadratic function $f(x)$ such that $f(2)=-20$ is its minimum value, and such that $f(4)=-8$.

Pr 8 Find an equation of the non-vertical line which intersects the parabola $y=x^2$ only at the point $(3,9)$.

Pr 9 Find the maximum value for the function $f(x)=\frac{12x}{x^2+4}$.

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