Tables and Graphs of Functions
Marcus is making a poster of the path of a basketball free throw. His calculator battery is almost dead, so he can only compute five points of the parabola. Which five points should he choose so the sketch comes out right? That is the skill of this page: choosing the right values of $x$, computing the outputs, and connecting the points with good judgment.
Building a Table of Values
A table of values lists inputs $x$ and the outputs $f(x)$. Each row is a point $(x, f(x))$ on the graph. Random inputs work for a line, but for most other functions you need to pick them on purpose.
For a line, two points determine the graph and a third checks your work.
For a parabola, the vertex and the intercepts come first.
Given the linear function $f(x) = 2x - 3$: a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$. b) Plot the function on the Cartesian plane using the points…
Given the linear function $f(x) = 2x - 3$:
- a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$.
- b) Plot the function on the Cartesian plane using the points obtained.
Consider the quadratic function $f(x) = x^2 - 4$: a) Calculate the images for $x = -3, -2, 0, 2, 3$ and organize the results in a table. b) Draw the graph of the…
Consider the quadratic function $f(x) = x^2 - 4$:
- a) Calculate the images for $x = -3, -2, 0, 2, 3$ and organize the results in a table.
- b) Draw the graph of the function and identify the coordinates of its vertex.
The cost $C(t)$ in dollars of leaving a car in a parking lot is given by the function $C(t) = 2 + 1.5t$, where $t$ is the time in hours. a) Make a table of values for…
The cost $C(t)$ in dollars of leaving a car in a parking lot is given by the function $C(t) = 2 + 1.5t$, where $t$ is the time in hours.
- a) Make a table of values for the first 4 hours ($t=0, 1, 2, 3, 4$).
- b) Plot the function in that interval.
From a Table to a Formula
Sometimes you get the table first and need the rule. Look at the differences between consecutive outputs, as long as the inputs go up by the same step each time.
Once you know the type, write the general form and use the points to find the unknown coefficients.
Identify whether the following table of values corresponds to a linear function or a quadratic function by calculating the differences between consecutive values of $y$.…
Identify whether the following table of values corresponds to a linear function or a quadratic function by calculating the differences between consecutive values of $y$. Then, find its analytical expression $y = f(x)$.
| $x$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|
| $y$ | $1$ | $4$ | $7$ | $10$ | $13$ |
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven since it is turned on is given by the function $T(t) = 20 + 15t$, where $t$ is measured in minutes. a)…
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven since it is turned on is given by the function $T(t) = 20 + 15t$, where $t$ is measured in minutes.
- a) What is the initial temperature of the oven?
- b) Construct a table of values for $t = 0, 5, 10, 15$.
- c) How much time must elapse for the oven to reach $200^\circ\text{C}$?
A quadratic function of the form $f(x) = ax^2 + bx + c$ passes through the points $(0, 3)$, $(1, 0)$ and $(2, -1)$. a) Set up and solve the system of equations to find…
A quadratic function of the form $f(x) = ax^2 + bx + c$ passes through the points $(0, 3)$, $(1, 0)$ and $(2, -1)$.
- a) Set up and solve the system of equations to find the values of $a, b$ and $c$.
- b) With the analytical expression obtained, calculate the vertex and construct a table of values to plot the parabola.
Parabolas: Vertex, Intercepts and Models
A quadratic function $f(x) = ax^2 + bx + c$ has a graph with a clear skeleton, so you can sketch it with very few points.
Quadratics also model real situations: the vertex gives the best (maximum or minimum) value, and the $x$-intercepts are where a quantity hits zero (a profit of $0$ is a break-even point).
When one side of a pen is a wall, that side needs no fence, so the fencing is used on only three sides. Write the missing side in terms of $x$ first, then the area.
Consider the quadratic function $f(x) = -x^2 + 6x - 5$. a) Calculate analytically the coordinates of its vertex and the points of intersection with the axes. b) Create a…
Consider the quadratic function $f(x) = -x^2 + 6x - 5$.
- a) Calculate analytically the coordinates of its vertex and the points of intersection with the axes.
- b) Create a table of values taking two points to the left of the vertex and two to the right.
- c) Plot the parabola.
The monthly profit $B(x)$, in dollars, of an electronic components factory as a function of the number of units produced $x$ is modeled by the function…
The monthly profit $B(x)$, in dollars, of an electronic components factory as a function of the number of units produced $x$ is modeled by the function $B(x) = -5x^2 + 300x - 4000$.
- a) Find the break-even points (production for which the profit is zero).
- b) Generate a strategic table of values that includes the break-even points and the vertex.
- c) How many units must be produced to maximize the profit and what is this maximum profit?
We want to build a rectangular pen using an existing wall for one of its sides. There are $20\text{ m}$ of metal fencing available for the other three sides. a) Express…
We want to build a rectangular pen using an existing wall for one of its sides. There are $20\text{ m}$ of metal fencing available for the other three sides.
- a) Express the area $A(x)$ of the pen as a function of the length $x$ of one of the sides perpendicular to the wall.
- b) Determine the domain of validity of the function $A(x)$ in the context of the problem.
- c) Construct a table of values and use it to sketch the graph and find the dimensions that maximize the area.
Radical Functions: Where the Graph Starts
A square root needs a nonnegative radicand, so the graph of a radical function starts exactly where the radicand is $0$. Some problems call these irrational functions; they are the same thing as radical functions.
When the radicand is $9 - x^2$ or $16 - x^2$, the inequality is a quadratic one. For $\sqrt{16 - x^2}$, we need $16 - x^2 \ge 0$, so $-4 \le x \le 4$. Squaring both sides of $y = \sqrt{16 - x^2}$ gives $x^2 + y^2 = 16$, a circle of radius $4$. The graph is the upper half of that circle (because $y \ge 0$), so the range is $[0, 4]$.
Given the irrational function $f(x) = \sqrt{x+2}$: a) Determine its domain analytically. b) Complete a table of values for $x \in \{-2, -1, 2, 7\}$ and use it to sketch…
Given the irrational function $f(x) = \sqrt{x+2}$:
- a) Determine its domain analytically.
- b) Complete a table of values for $x \in \{-2, -1, 2, 7\}$ and use it to sketch the graph.
Consider the function $f(x) = \sqrt{9-x^2}$. a) Calculate its domain of definition analytically. b) Make a table of values for $x \in \{-3, -2, 0, 2, 3\}$. c) Plot the…
Consider the function $f(x) = \sqrt{9-x^2}$.
- a) Calculate its domain of definition analytically.
- b) Make a table of values for $x \in \{-3, -2, 0, 2, 3\}$.
- c) Plot the function and state its range.
Absolute Value Functions
$|x|$ is the distance from $x$ to $0$, so it is never negative. In $y = |g(x)|$, any part of the graph of $g$ that falls below the $x$-axis gets flipped up.
Given the absolute value function $f(x) = |2x - 5|$: a) Evaluate the function at the points $x = 0, 2, 2.5, 3, 5$ and construct a table of values. b) Plot the function.
Given the absolute value function $f(x) = |2x - 5|$:
- a) Evaluate the function at the points $x = 0, 2, 2.5, 3, 5$ and construct a table of values.
- b) Plot the function.
Given the absolute value function $f(x) = |x^2 - 4|$: a) Express the piecewise-defined function by removing the absolute value bars. b) Find the points of intersection…
Given the absolute value function $f(x) = |x^2 - 4|$:
- a) Express the piecewise-defined function by removing the absolute value bars.
- b) Find the points of intersection with the axes.
- c) Construct a table of values in the interval $[-3, 3]$ and draw its graph.
Rational Functions: Points Near the Forbidden Value
A rational function has a variable in the denominator, so you must exclude the $x$-values that make the denominator $0$. The interesting behavior happens right next to those values.
Calculate analytically the points of intersection with the x-axis and the y-axis of the rational function $f(x) = \frac{3x+6}{x-1}$.
Calculate analytically the points of intersection with the x-axis and the y-axis of the rational function $f(x) = \frac{3x+6}{x-1}$.
Given the rational function $f(x) = \frac{4}{x-2}$: a) Determine its domain. b) Construct a table of values by evaluating the function at points close to $x = 2$ from…
Given the rational function $f(x) = \frac{4}{x-2}$:
- a) Determine its domain.
- b) Construct a table of values by evaluating the function at points close to $x = 2$ from the left and from the right ($x = 1.9, 1.99, 2.1, 2.01$).
- c) From the results obtained, what graphical feature does the function have at $x = 2$?
Piecewise Functions: Checking the Break
A piecewise function uses a different rule on different intervals. Check each pair of pieces at the point where the rule changes.
A function is continuous at the break $x = c$ when both rules give the same value there. This idea lets you find an unknown parameter.
Given the piecewise-defined function: $f(x) = \begin{cases} -x + 2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}$ a) Construct a table of values by…
Given the piecewise-defined function:
$f(x) = \begin{cases} -x + 2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}$
- a) Construct a table of values by evaluating the function at $x = -2, 0, 0.99, 1, 2, 3$.
- b) Plot the function and indicate whether it is continuous at $x = 1$.
Given the piecewise function depending on a parameter $k$: $f(x) = \begin{cases} x^2 - k & \text{if } x < 2 \\ 2x + 1 & \text{if } x \ge 2 \end{cases}$ a) Find the value…
Given the piecewise function depending on a parameter $k$:
$f(x) = \begin{cases} x^2 - k & \text{if } x < 2 \\ 2x + 1 & \text{if } x \ge 2 \end{cases}$
- a) Find the value of $k$ so that the two branches of the function "connect" at the point $x = 2$ (that is, so that the function is continuous).
- b) With the value of $k$ found, construct a table of values in the interval $[0, 4]$ and sketch the graph of the function.
Exponential and Logarithmic Functions
In an exponential function $y = a \cdot b^x$ the input is in the exponent. Each time $x$ goes up by $1$, the output is multiplied by the growth factor $b$ (rather than increased by a fixed amount).
The logarithmic function is the reverse of an exponential. $y = \log_2 x$ means $2^y = x$. The argument of a logarithm must be positive, so the domain comes from "argument $> 0$", and the line where the argument is $0$ is a vertical asymptote.
In a laboratory, the growth of a population of bacteria is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \cdot 2^t$, where $t$…
In a laboratory, the growth of a population of bacteria is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \cdot 2^t$, where $t$ is the time in hours.
- a) Construct a table of values for the first 4 hours ($t = 0, 1, 2, 3, 4$).
- b) Plot the function. What type of growth is observed?
The mass $M(t)$ in grams of a radioactive isotope decreases with time $t$ (in days) according to the exponential function $M(t) = M_0 \cdot (0.5)^{\frac{t}{T}}$, where…
The mass $M(t)$ in grams of a radioactive isotope decreases with time $t$ (in days) according to the exponential function $M(t) = M_0 \cdot (0.5)^{\frac{t}{T}}$, where $M_0$ is the initial mass and $T$ is the half-life.
Knowing that at time $t=0$ there were $200\text{ g}$ of substance and that after $60\text{ days}$ there are $50\text{ g}$ left:
- a) Calculate the value of the parameters $M_0$ and $T$.
- b) Construct a table of values for $t = 0, 30, 60, 90, 120$ and sketch the graph.
Given the logarithmic function $f(x) = \log_2(x+4)$: a) Determine analytically its domain and the equation of its vertical asymptote. b) Construct a table of values by…
Given the logarithmic function $f(x) = \log_2(x+4)$:
- a) Determine analytically its domain and the equation of its vertical asymptote.
- b) Construct a table of values by choosing strategic values of $x$ that result in exact powers of base 2 for the argument of the logarithm.
- c) Plot the graph of the function.
Tables for an Inverse Function
If $f$ sends $x$ to $y$, its inverse $f^{-1}$ sends $y$ back to $x$. So you get a table of $f^{-1}$ by swapping the columns of the table of $f$.
The following figure shows the graph of the exponential function $f(x) = 2^x$ and the line $y = x$. a) From the highlighted points on the blue curve of $f(x)$, construct…
The following figure shows the graph of the exponential function $f(x) = 2^x$ and the line $y = x$.
- a) From the highlighted points on the blue curve of $f(x)$, construct a table of values for its inverse function $f^{-1}(x)$.
- b) Draw the graph of the inverse function in your notebook, taking advantage of the symmetry with respect to the line $y=x$.
- c) Write the analytical expression of $f^{-1}(x)$.
Reading a Graph: Domain, Range, Trends and Asymptotes
Sometimes the graph is already drawn and you need to read it. Look at it in this order: which $x$-values are used (domain), which $y$-values are reached (range), special points, whether it goes up or down, and what happens at the edges.
Graphs of real situations often level off. A hot drink cools quickly at first, then more and more slowly, and ends up at the temperature of the room.
You can also go backward from a graph to a formula. A rational function $f(x) = \dfrac{ax + b}{cx + d}$ has vertical asymptote $x = -\dfrac{d}{c}$ and horizontal asymptote $y = \dfrac{a}{c}$.
Observe the following graph of a function $f(x)$ defined by straight line segments: From the graph, determine: a) The domain and the range of the function. b) The value…
Observe the following graph of a function $f(x)$ defined by straight line segments:
From the graph, determine:
- a) The domain and the range of the function.
- b) The value of $f(-2)$ and $f(1)$.
The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of the time $t$ (in minutes) since it is served.…
The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of the time $t$ (in minutes) since it is served.
From the graph:
- a) State the initial temperature of the coffee.
- b) What value does the temperature tend towards as a lot of time passes? What is the physical meaning of the horizontal asymptote?
- c) Is the function increasing or decreasing? Justify your answer.
Observe the following graph of a rational function of the form $f(x) = \frac{ax+b}{cx+d}$: From the information provided by the graph (asymptotes and points of…
Observe the following graph of a rational function of the form $f(x) = \frac{ax+b}{cx+d}$:
From the information provided by the graph (asymptotes and points of intersection):
- a) Determine the domain and the range of the function.
- b) Deduce the analytical expression of the function $f(x)$.
Back to the start: for Marcus's free throw, the five best points are the vertex (the top of the shot), the two $x$-intercepts where the ball is at ground level (or the release height), and one point on each side of the vertex to confirm the symmetry. That is the same strategy you used for every function on this page.