A Generic Quadratic Function: $f(x)=ax^2+bx+c$.
Fact: Every quadratic function is the shape of a parabola.
Important Features of a Quadratic Function
1) $y$-intercept
2) $x$-intercept(s)
3) Vertex.
4) Opens Up or Down
Important Features of a Quadratic Function
For the function $f(x)=ax^2+bx+c$
1) The $y$-intercept is $(0,c).$
2) To find the $x$-intercept(s), solve $ax^2+bx+c=0.$
3) Vertex. The $x$-coordinate is $-\frac{b}{2a}$. The $y$-coordinate is $f(-\frac{b}{2a}).$
4) Opens Up or Down: If $a$ is positive, the parabola opens up. If $a$ is negative, the parabola opens down.
Application: wants to build a rectangular pen for some prize-winning peafowl against the side of a barn. With a total of 66 feet of fencing, find the length and width which maximize the area of the pen.
Application: Billy Bob owns a coal mine. One day he noticed, contrary to his expectations, that selling more coal doesn't necessarily mean more profit. Looking at data from the last year, Billy Bob created a function $P(x)=−80x^2+7000x−50000$ which gives the profit in dollars when $x$ tons of coal are produced and sold. Graph this function and use it to help Billy Bob determine the following:
a. his overhead costs (Hint: Evaluate $P(0).$)
b. the break-even values (Hint: When does $P(x)=0?$) Round your answers to the nearest hundredth.
c. his maximum profit that can be made and the number of tons to sell to create this profit (find the vertex).