Tutorial: Quadratic functions and models

(This topic is also in Section 2.1 in Applied Calculus or Section 2.1 in Finite Mathematics and Applied Calculus)

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Tutorial:

Chapter true/false quiz

Brush up on your algebra For all of these tutorials on nonlinear functions, you should be familiar with the %%algexponensTut Also, you should review the %%solvingquadraticsTut.

Fundamentals

The relationship between two quantities is often best modeled by a curved line rather than a straight line. The simplest function whose graph that is not straight line is a quadratic function.

Quadratic function
A quadratic function of the variable $x$ is a function that can be written in the form

$f(x) = ax^2 + bx + c$ \gap[40] \t Function form \\ $y = ax^2 + bx + c$ \gap[40] \t Equation form

where $a, b,$ and $c$ are fixed numbers (with $a \neq 0$).

%%Examples
#[The following matrices are reduced:][Las siguientes matrices son redicidas:]#

1. \t $\displaystyle f(x) = 3x^2-2x+1$ \t $a = 3, b = -2, c = 1$ \\ 2. \t $\displaystyle g(x) = -\frac{x^2}{2}$ \t $\displaystyle a = -\frac{1}{2}, b = c = 0$ \\ 3. \t $\displaystyle g(x) = 3x+1$ \t Not a quadratic function because $a = 0.$

%%YourTurn

Graph of a quadratic function

The graph of a quadratic function is a parabola (see the figure below).

Concavity: If the coefficient $a$ of $x^2$ is positive, it is concave up (as in the figure below when you press "$a \gt 0$"). If $a$ is negative, it is concave down (as in the figure below when you press "$a \lt 0$").

Vertex (Press the "Vertex" button above):The vertex of this parabola occurs at the point on the graph with

#[Vertex][Vértice]#

y-Intercept: (Press the "$y$-intercept" button above) The graph crosses the $y$-axis at

#[y-Intercept][Intersección en y]#

x-Intercept(s): (Press the "$x$-intercept" button above) It crosses the $x$-axis at the solution(s) of the quadratic equation $ax^2+bx+c = 0$ (if there are any). We can sometimes solve this equation by factoring the left-hand side. If it does not factor, we can always use the quadratic formula: Click here to review solving quadratic equations.

#[x-Intercept(s)][Intersección(es) en x]#

Note that, if the discriminant $b^2-4ac$ is negative, then there are no $x$-intercepts as then the quadratic equation has no real solutions.

Symmetry: It is symmetric around the vertical line through the vertex.

Example

%%Let $f(x) = -3x^2-6x-3.$ %%Therefore $a = -3, \ b = -6,\ c = -3.$

Concavity: As $a=-3$ is negative, the graph is concave down (press the "a < 0" button above to see its shape).

Vertex: (press the "Vertex" button above) The $x$-coordinate is

$x = -\dfrac{b}{2a} = -\dfrac{(-6)}{2(-3)} = -1$

The corresponding $y$-coordinate is

$y = f(-1) = -3(-1)^2-6(-1)-3 = -3+6-3 = 0$.

So, the vertex is at $(-1,0)$: y-Intercept: It crosses the $y$-axis at $y = c = -3$. x-Intercept(s): It crosses the $x$-axis at the solution(s) of $ax^2+bx+c = 0$:

$-3x^2-6x-3 = 0$ \\ $\implies -3(x^2+2x+1) = 0$ \\ $\implies -3(x+1)^2 = 0$ \gap[4] Luckily, the left-hand side factors. \\ $\implies x = -1$. \gap[4] Only one $x$-intercept, as already seen on the graph.

Applications

Recall that the revenue resulting from one or more transactions is the total payment received. (See the tutorial on functions and models.) Thus, if $q$ units of some item are sold at $p$ dollars per unit, the revenue resulting from the sale is

Now try the exercises in Section 2.1 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.