Marcus is coding a basketball game. For every shot he needs three things: how high the ball starts, how high it peaks, and where it lands. He could plot 50 points, or he could read all three facts straight from the formula.
Analyzing a function means collecting facts like these in an organized way, so you can sketch the whole graph without plotting dozens of points. By the end of this page you'll have a five-step checklist, and you'll use it on Marcus's shot.
The checklist and reading a graph
Domain and range from a graph
A closed dot means that endpoint is included. An open dot means the graph gets close to it but never reaches it, so it is not included. You can write answers as inequalities or in interval notation: $-3 \le x < 4$ is the same as $[-3, 4)$.
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Look at the following graph and determine its domain and range.
Look at the following graph and determine its domain and range.
When a graph models a real situation, the domain and range are limited by the context. A bacteria population measured for 10 hours has domain $0 \le t \le 10$ (time can't be negative), and its range starts at the starting population. A flat piece of the graph means the population is constant during that time interval.
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From the following graph, determine the intervals where the function is increasing and decreasing, and find the coordinates of its relative extrema.
From the following graph, determine the intervals where the function is increasing and decreasing, and find the coordinates of its relative extrema.
The following graph shows the growth of a bacterial population (in thousands) cultured in a laboratory over time (in hours). Analyze the domain and range in the context…
The following graph shows the growth of a bacterial population (in thousands) cultured in a laboratory over time (in hours). Analyze the domain and range in the context of the problem, and indicate on which intervals the population is increasing, decreasing, or constant.
Piecewise functions, absolute value, and continuity
A piecewise function uses different formulas on different parts of its domain. To evaluate it, first decide which piece the input belongs to.
Absolute value as a piecewise function
$|u| = u$ when $u \ge 0$ and $|u| = -u$ when $u < 0$. To rewrite $|\text{expression}|$ as a piecewise function, find where the inside equals $0$ and split there.
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Analyze the continuity of the following piecewise function at the point $x = 1$:…
Analyze the continuity of the following piecewise function at the point $x = 1$:
$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$
Given the following piecewise function, analyze its domain, its continuity, its x- and y-intercepts, and the intervals where it is increasing and decreasing:…
Given the following piecewise function, analyze its domain, its continuity, its x- and y-intercepts, and the intervals where it is increasing and decreasing:
Two graphs can both be increasing and still look very different: one curves like a smile, the other like a frown.
For parabolas it is easy: $y = ax^2 + bx + c$ is concave up if $a > 0$ and concave down if $a < 0$. For a piecewise function made of parabolas, check each piece.
Notice the sharp corners at $x = -3$ and $x = 3$: the graph changes concavity there, so those points are the boundaries of the intervals and are not included in them.
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From the quadratic function $f(x) = x^2 - 4x$, obtain the analytical expression as a piecewise function of $g(x) = |x^2 - 4x|$. Visualize the situation and determine the…
From the quadratic function $f(x) = x^2 - 4x$, obtain the analytical expression as a piecewise function of $g(x) = |x^2 - 4x|$. Visualize the situation and determine the intervals where $g(x)$ is concave up and concave down.
Some graphs get closer and closer to a line without touching it. That line is an asymptote. Why does it happen? Look at $f(x) = \dfrac{1}{x - 2}$.
$x$
$2.1$
$2.01$
$2.001$
$1.99$
$\dfrac{1}{x-2}$
$10$
$100$
$1{,}000$
$-100$
As $x$ gets close to $2$, we divide $1$ by a tiny number, and the output explodes. That is a vertical asymptote at $x = 2$.
Now go the other way. For $x = 10, 100, 1000$ we get $0.125$, about $0.0102$, about $0.001$. As $x$ gets larger and larger (written $x \to \infty$), the output gets closer to $0$. That is a horizontal asymptote at $y = 0$.
When the conditions are harder to read off, write each one as an equation in $a$, $b$, $c$ and solve the resulting system.
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Find the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{5}{x - 3} + 2$.
Find the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{5}{x - 3} + 2$.
The rational function $f(x) = \frac{ax + b}{x + c}$ passes through the point $(0, 1)$, has a vertical asymptote at the line $x = 2$, and has a horizontal asymptote at…
The rational function $f(x) = \frac{ax + b}{x + c}$ passes through the point $(0, 1)$, has a vertical asymptote at the line $x = 2$, and has a horizontal asymptote at the line $y = 3$. Set up a system of equations and find the values of the parameters $a$, $b$, and $c$.
The position of an object in linear motion is given by the function $e(t) = 5t^2 + 2t$, where $e$ is the distance in feet and $t$ is the time in seconds. Calculate the…
The position of an object in linear motion is given by the function $e(t) = 5t^2 + 2t$, where $e$ is the distance in feet and $t$ is the time in seconds. Calculate the average speed (average rate of change) on the intervals $[1, 3]$ and $[3, 5]$, and indicate in which interval the object moved faster.
The concentration of a certain drug in a patient's blood, in mg/L, $t$ hours after its administration is given by the model $C(t) = \frac{10t}{t^2 + 1}$. Calculate the…
The concentration of a certain drug in a patient's blood, in mg/L, $t$ hours after its administration is given by the model $C(t) = \frac{10t}{t^2 + 1}$. Calculate the average rate of change over the first hour (interval $[0, 1]$) and interpret the result. Furthermore, explain mathematically what the long-term trend of the drug's concentration is (as $t \to \infty$).
When a function models a real situation, your checklist answers real questions. For a quadratic $f(x) = ax^2 + bx + c$:
Back to Marcus's shot: this is exactly what he needed. The starting height is the y-intercept, the peak is the vertex, and the landing time is the positive zero. He got all three from the formula, with no plotting.
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The profit of a startup company, in thousands of dollars, is given by the function $B(x) = -0.5x^2 + 10x - 18$, where $x$ represents the years elapsed since its…
The profit of a startup company, in thousands of dollars, is given by the function $B(x) = -0.5x^2 + 10x - 18$, where $x$ represents the years elapsed since its founding. Determine in which year the maximum profit is reached, how much this profit is, and in which years the company breaks even (zero profit).
A projectile is launched upward and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in feet and $t$ is the time in…
A projectile is launched upward and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in feet and $t$ is the time in seconds since launch. Determine the domain of the function restricted to the actual duration of the flight (until it hits the ground), the height from which it was launched, the maximum height it reaches, and the total flight time.
In an isosceles triangle with a base of $12$ in and a height of $10$ in, a rectangle is inscribed whose base $x$ rests on the base of the triangle, as shown in the…
In an isosceles triangle with a base of $12$ in and a height of $10$ in, a rectangle is inscribed whose base $x$ rests on the base of the triangle, as shown in the figure. Express the area of the rectangle as a function of its base $x$ and determine the domain of this function in the geometric context of the problem.
To see the shape, rewrite $f(x) = 1 + \dfrac{3}{x^2 - 4}$. For $x > 2$, the fraction is positive and shrinks as $x$ grows, so $f$ decreases from very large values toward $1$. For $0 \le x < 2$ the fraction is negative and plunges to $-\infty$ as $x \to 2$, so $f$ decreases from $\tfrac14$. By evenness, $f$ is increasing on $(-\infty, -2)$ and $(-2, 0)$, decreasing on $(0, 2)$ and $(2, \infty)$, with a relative maximum at $\left(0, \tfrac14\right)$.
Sketching from properties. To draw a graph from a list of conditions, first draw the asymptotes as dashed lines. Then mark the given points and mirror them if the function is even or odd. Finally, connect the points with curves that respect the increasing/decreasing behavior and approach the asymptotes.
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Describe the behavior of the function $f(x) = \frac{2x^2}{x^2 + 1}$ by analyzing its domain, its x- and y-intercepts, its symmetry, and its horizontal and vertical…
Describe the behavior of the function $f(x) = \frac{2x^2}{x^2 + 1}$ by analyzing its domain, its x- and y-intercepts, its symmetry, and its horizontal and vertical asymptotes.
Prove analytically whether the function $f(x) = \frac{x^3}{x^2 - 4}$ is even, odd, or neither. Then, find its x- and y-intercepts and determine the equations of its…
Prove analytically whether the function $f(x) = \frac{x^3}{x^2 - 4}$ is even, odd, or neither. Then, find its x- and y-intercepts and determine the equations of its vertical and horizontal asymptotes.
Sketch the graph of a function $f(x)$ that simultaneously satisfies all the following conditions: Its domain is $\mathbb{R} \setminus \{-2, 2\}$. It is an odd function.…
Sketch the graph of a function $f(x)$ that simultaneously satisfies all the following conditions:
Its domain is $\mathbb{R} \setminus \{-2, 2\}$.
It is an odd function.
It has vertical asymptotes at $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 0$.
It has a relative maximum at the point $(-4, 1)$ and a relative minimum at $(4, -1)$.