Tables and Graphs of Functions
Your teacher says: "Sketch $f(x) = x^2 - 6x + 8$ on the quiz. No Desmos." You could plug in random $x$-values and hope for the best. But which values will actually show you the shape of the graph? Choosing the right points is half the work of graphing.
Building a Table: Which Points Matter?
A table of values pairs inputs with outputs: for each $x$ you compute $f(x)$ and plot the point $(x, f(x))$. Some people call $f(a)$ the image of $a$. It is just the output.
For a linear function $f(x) = mx + b$, two points determine the line. Use a third as a check. Always include $x = 0$, which gives the $y$-intercept $b$.
For a quadratic $f(x) = ax^2 + bx + c$, the graph is a parabola that is symmetric about the vertical line through its vertex. Find the vertex, then take points on each side.
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.
Consider the quadratic function $f(x) = -x^2 + 6x - 5$. a) Calculate analytically the coordinates of its vertex and the x- and y-intercepts. b) Create a table of values…
Consider the quadratic function $f(x) = -x^2 + 6x - 5$.
- a) Calculate analytically the coordinates of its vertex and the x- and y-intercepts.
- b) Create a table of values taking two points to the left of the vertex and two to the right.
- c) Plot the parabola.
Linear Models and Differences in a Table
Real situations with a fixed starting amount and a constant rate are linear. A gym charges a $\$20$ sign-up fee plus $\$15$ per month: $C(m) = 20 + 15m$. The slope $15$ is the rate of change, and the $y$-intercept $20$ is the starting value.
You can also recognize the type of function from a table, when $x$ goes up by the same step each time.
Once you have a formula, you can answer questions like "when does the cost reach $\$155$?" by setting the formula equal to $155$: $20 + 15m = 155$, so $15m = 135$ and $m = 9$ months.
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 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.
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 $200^\circ\text{C}$?
Quadratic Models and Their Maximum
Quadratic formulas model a ball in flight, a profit, or the area of a rectangle with a fixed amount of fence. In each case the vertex answers the question "what is the maximum (or minimum)?"
Finding a quadratic from three points
If you know three points, substitute each into $y = ax^2 + bx + c$ to get a system of three equations, then solve it with elimination.
Domain in context
A parabola goes on forever, but a real situation does not. A rectangular garden with $40$ ft of fence has width $w$ and length $20 - w$, so $A(w) = w(20 - w) = -w^2 + 20w$. The vertex is at $w = 10$ with $A = 100$ ft², but only $0 < w < 20$ makes sense, because both sides must be positive. Keep only that part of the graph.
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?
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.
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
For $f(x) = \sqrt{\ \ }$ of an expression, the radicand must be at least $0$ (you can't take a real square root of a negative number). Solve "radicand $\ge 0$" to get the domain, and start your table at the endpoint.
Expressions like $\sqrt{r^2 - x^2}$ give a semicircle. For $h(x) = \sqrt{4 - x^2}$, solve $4 - x^2 \ge 0$: the domain is $[-2, 2]$. Then $h(-2) = 0$, $h(0) = 2$, $h(2) = 0$, and $h(\pm 1) = \sqrt{3} \approx 1.73$. The graph is the upper half of a circle of radius $2$, so the range is $[0, 2]$.
Given the radical 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…
Given the radical 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 analytically. b) Make a table of values for $x \in \{-3, -2, 0, 2, 3\}$. c) Plot the function and…
Consider the function $f(x) = \sqrt{9-x^2}$.
- a) Calculate its domain 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
The absolute value $|u|$ is the distance from $u$ to $0$. As a piecewise definition, $|u| = u$ if $u \ge 0$, and $|u| = -u$ if $u < 0$. So the graph of $y = |\,\text{expression}\,|$ is the graph of the expression with every negative part flipped above the $x$-axis.
The graph has a corner wherever the expression inside equals $0$, so always include that point in your table.
When the inside is quadratic, solve for where it is positive or negative. For $h(x) = |x^2 - 1|$: the inside $x^2 - 1 \ge 0$ when $x \le -1$ or $x \ge 1$. So
$$h(x) = \begin{cases} x^2 - 1 & \text{if } x \le -1 \text{ or } x \ge 1 \\ -x^2 + 1 & \text{if } -1 < x < 1 \end{cases}$$The $x$-intercepts are $\pm 1$ and the $y$-intercept is $h(0) = 1$.
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 x- and y-intercepts. c)…
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 x- and y-intercepts.
- c) Construct a table of values in the interval $[-3, 3]$ and draw its graph.
Rational Functions: Intercepts and Forbidden Values
For a rational function $f(x) = \dfrac{\text{numerator}}{\text{denominator}}$:
Near a forbidden value, the outputs get huge. That is a vertical asymptote. As $|x|$ gets huge, the outputs settle near a height, a horizontal asymptote. Neither is part of the graph; they are guidelines.
Find the x- and y-intercepts of the rational function $f(x) = \frac{3x+6}{x-1}$ analytically.
Find the x- and y-intercepts of the rational function $f(x) = \frac{3x+6}{x-1}$ analytically.
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 and Continuity
On a graph, a point that belongs to a piece is a closed circle. A point that the piece approaches but doesn't include is an open circle. A function is continuous at a break if you can draw through it without lifting your pencil: both pieces must meet at the same point.
Making a function continuous
To connect two pieces at $x = a$, set their values at $a$ equal and solve for the unknown.
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 Functions
An exponential function has the form $f(x) = a \cdot b^x$ with $b > 0$ and $b \ne 1$. The number $a$ is the $y$-intercept (the value at $x = 0$), and $b$ is the growth factor.
-
If $b > 1$ the function shows growth.
-
If $0 < b < 1$ the function shows decay.
The $x$-axis ($y = 0$) is a horizontal asymptote.
Linear functions add the same amount each step; exponential functions multiply by the same factor each step. In a table, constant ratios $\dfrac{y_{\text{next}}}{y}$ signal an exponential.
Half-life
Radioactive decay is written $M(t) = M_0 \cdot (0.5)^{t/T}$, where $M_0$ is the starting amount and $T$ is the half-life, the time it takes for half of the amount to disappear. The exponent $t/T$ counts how many half-lives have passed. For example, with $80$ oz and $T = 10$ days, the amounts at $t = 0, 10, 20, 30$ are $80, 40, 20, 10$ oz.
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 ounces 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 ounces 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 $200\text{ oz}$ of substance and that after $60\text{ days}$ there are $50\text{ oz}$ 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.
Logarithmic Functions and Inverses
The function $y = \log_b x$ answers the question "what exponent gives $x$ from base $b$?" That is, $y = \log_b x$ means $b^y = x$. Logarithms undo exponentials.
Graphing a logarithm
For $\log_b(\text{expression})$, the argument must be positive. Setting it equal to $0$ gives the vertical asymptote. To build a table, choose $x$ so the argument is a power of $b$: $\ldots, \frac{1}{2}, 1, 2, 4, 8, \ldots$ for base $2$.
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.
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 Information from a Graph
You won't always build a graph. Often you must read one that is already drawn. Use this checklist:
| Question | Where to look |
|---|---|
| Domain | the $x$-values the graph covers (left to right) |
| Range | the $y$-values the graph covers (bottom to top) |
| $f(a)$ | go to $x = a$, then read the height |
| Intercepts | where the graph meets the axes |
| Increasing / decreasing | going left to right, does the graph rise or fall? |
| Asymptotes | lines the graph gets closer and closer to (a horizontal one can sometimes be crossed) |
Trends and asymptotes
A graph can level off. Suppose coffee at $190$ °F cools in a $70$ °F room.
The graph is decreasing and flattens out: as time passes, the temperature approaches $70$ °F. The line $y = 70$ is a horizontal asymptote. Its meaning: the coffee cools toward room temperature and never goes below it. In a model like this one, the graph gets closer and closer to $y = 70$ without reaching it.
Writing a formula from a rational graph
For $f(x) = \dfrac{ax + b}{cx + d}$, the graph tells you:
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vertical asymptote at $x = -\dfrac{d}{c}$ (denominator is $0$),
-
horizontal asymptote at $y = \dfrac{a}{c}$ (ratio of leading coefficients),
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$x$-intercept at $x = -\dfrac{b}{a}$ (numerator is $0$).
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 intercepts): a)…
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 intercepts):
- a) Determine the domain and the range of the function.
- b) Deduce the analytical expression of the function $f(x)$.
Summary
Back to the opening: to sketch $x^2 - 6x + 8$ without Desmos, find the vertex $(3, -1)$, the intercepts $(0, 8)$, $(2, 0)$, $(4, 0)$, and a point on each side, then draw a smooth curve.