Table and Graph of a Function
A parking lot charges a flat fee plus an amount per hour, and Ximena wants to see at a glance how much she will pay depending on how long she leaves her car. For that you need to know how to graph: choose good values of $x$, calculate their outputs, and connect the points sensibly. How do you pick the points that are really worth it? We'll find out on this page.
Table of values: which points to choose
A table of values pairs each $x$ with its output $f(x)$. Each pair $(x, f(x))$ is a point on the graph in the coordinate plane.
For a linear function $f(x)=mx+b$ two points are enough, but a third one is useful as a check. The y-intercept is $f(0)=b$, and the x-intercept comes from solving $mx+b=0$.
For a quadratic $f(x)=ax^2+bx+c$, the vertex is at $x_v=-\dfrac{b}{2a}$. The x-intercepts are obtained with the quadratic formula, and the y-intercept is $c$. Then you take points on both sides of the vertex: the parabola is symmetric about the line $x=x_v$.
When the function is exponential or logarithmic, it is best to choose $x$ values that give exact results. For example, for $N(t)=100\cdot2^t$ you take $t=0,1,2,3,4$.
Logarithmic function: domain and vertical asymptote
The logarithm only exists for positive arguments: $\log_2(u)$ requires $u>0$. That is why the domain of $f(x)=\log_2(x+4)$ comes from $x+4>0$, that is, $x>-4$, or $(-4,\infty)$.
When $x$ approaches $-4$ from the right, the argument $x+4$ approaches $0^+$ and $\log_2(x+4)$ decreases without bound. That is why the line $x=-4$ is a vertical asymptote: the graph gets closer and closer to it without touching it.
For the table you choose $x$ such that $x+4=\tfrac12,1,2,4,8$, that is, $x=-3.5,-3,-2,0,4$, and so the logarithms equal $-1,0,1,2,3$.
Given the linear function $f(x) = 2x - 3$: a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$. b) Draw the graph of the function in the Cartesian plane using the…
Given the linear function $f(x) = 2x - 3$:
- a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$.
- b) Draw the graph of the function in the Cartesian plane using the points obtained.
You can do it on paper and send a photo with your answer.
Consider the quadratic function $f(x) = x^2 - 4$: a) Calculate the images for $x = -3, -2, 0, 2, 3$ and organise 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 organise the results in a table.
- b) Draw the graph of the function and identify the coordinates of its vertex.
You can do it on paper and send a photo with your answer.
Consider the quadratic function $f(x) = -x^2 + 6x - 5$. a) Calculate analytically the coordinates of its vertex and its axis 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 its axis intercepts.
- b) Create a table of values taking two points to the left of the vertex and two to the right.
- c) Draw the graph of the parabola.
You can do it on paper and send a photo with your answer.
Is this graph a function? The vertical line test
A function gives exactly one output for each $x$. On a graph this looks like this: if you draw a vertical line at any position, it must cross the curve at most once.
Tables to detect the type of function
If the values of $x$ go up by one, the differences between consecutive values of $y$ tell you what type of function it is.
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If the first difference is constant, the function is linear and that difference is the slope $m$.
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If the first difference changes but the second is constant, it is quadratic.
For example, with $y=1,4,7,10,13$ the differences are $3,3,3,3$: it is linear with $m=3$. Since $f(0)=4$, we get $y=3x+4$.
When three points of a quadratic $ax^2+bx+c$ are known, each point gives an equation and you solve the $3\times3$ system in $a$, $b$, and $c$. Since one point is usually the y-intercept, $c$ is obtained directly.
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 algebraic expression $y = f(x)$.
| $x$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|
| $y$ | $1$ | $4$ | $7$ | $10$ | $13$ |
The cost $C(t)$ in pesos of leaving a car in a parking lot is given by the function $C(t) = 40 + 30t$, where $t$ is the time in hours. a) Make a table of values for the…
The cost $C(t)$ in pesos of leaving a car in a parking lot is given by the function $C(t) = 40 + 30t$, 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) Draw the graph of the function in that interval.
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) Using the algebraic expression obtained, calculate the vertex and construct a table of values to plot the parabola.
You can do it on paper and send a photo with your answer.
Square root and rational functions: points near the excluded value
In a rational function, the interesting values are near the value that makes the denominator zero. In a square root function, the domain starts exactly where the radicand equals zero.
For the intercepts of a rational function $\dfrac{3x+6}{x-1}$: for the x-axis you set the numerator equal to zero ($x=-2$, as long as the denominator is not zero there), and for the y-axis you calculate $f(0)=\dfrac{6}{-1}=-6$.
For $f(x)=\sqrt{x+2}$ the radicand must satisfy $x+2\ge0$, that is, $x\ge-2$. The table starts at $x=-2$ and uses values that give exact roots ($x=-1,2,7$ give $1,2,3$).
For $f(x)=\sqrt{9-x^2}$ we require $9-x^2\ge0$, that is, $-3\le x\le3$. The points $(\pm3,0)$, $(0,3)$, and $(\pm2,\sqrt5)$ trace a semicircle, and the range is $[0,3]$.
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.
Calculate analytically the points of intersection with the x-axis and with 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 with 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 evaluating the function at points close to $x = 2$ from the…
Given the rational function $f(x) = \frac{4}{x-2}$:
- a) Determine its domain.
- b) Construct a table of values 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) Based on the results obtained, what graphical feature does the function display at $x = 2$?
You can do it on paper and send a photo with your answer.
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) Draw 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) Draw the graph of the function and state its range.
You can do it on paper and send a photo with your answer.
Piecewise-defined and absolute value functions
The absolute value is a piecewise-defined function: $|u|=u$ if $u\ge0$ and $|u|=-u$ if $u<0$. For $f(x)=|2x-5|$ the change of piece happens when $2x-5=0$, that is $x=2.5$, which is the vertex of the "V". For $f(x)=|x^2-4|$, the parabola is kept where $x^2-4\ge0$ (if $x\le-2$ or $x\ge2$) and is reflected upward between $-2$ and $2$:
$$f(x)=\begin{cases} x^2-4 & \text{if } x\le-2 \text{ or } x\ge2 \\ -x^2+4 & \text{if } -2Given the modulus 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) Draw the graph of the…
Given the modulus 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) Draw the graph of the function.
You can do it on paper and send a photo with your answer.
Given the piecewise 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…
Given the piecewise 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) Draw the graph of the function and state whether it is continuous at $x = 1$.
You can do it on paper and send a photo with your answer.
Given the absolute value function $f(x) = |x^2 - 4|$: a) Express the piecewise function by removing the absolute value. b) Find the axis intercepts. c) Construct a table…
Given the absolute value function $f(x) = |x^2 - 4|$:
- a) Express the piecewise function by removing the absolute value.
- b) Find the axis intercepts.
- c) Construct a table of values in the interval $[-3, 3]$ and draw its graph.
You can do it on paper and send a photo with your answer.
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) Using the value of $k$ found, construct a table of values in the interval $[0, 4]$ and sketch the graph of the function.
You can do it on paper and send a photo with your answer.
Models: graphs in real situations
Many problems give the function in a context. The table and the graph help with interpretation: $f(0)$ is the initial value, the slope is the rate of change, and the vertex of a parabola gives a maximum or a minimum.
Maximum profit. If $B(x)=-100x^2+6{,}000x-80{,}000$ (in pesos), the break-even points come from $B(x)=0$. Dividing by $-100$: $x^2-60x+800=0$, with $x=20$ and $x=40$. The vertex is at $x=\dfrac{-6000}{2(-100)}=30$ and $B(30)=-90{,}000+180{,}000-80{,}000=\$10{,}000$.
Exponential growth and decay. In $N(t)=100\cdot2^t$ the quantity doubles every hour: $100, 200, 400, 800, 1{,}600$. The graph rises faster and faster (exponential growth). In $M(t)=M_0\cdot0.5^{t/T}$, $M_0$ is the mass at $t=0$ and $T$ is the time in which the mass is reduced to half. With $M_0=200$ g and $50$ g after $60$ days, the mass was reduced to a quarter, which is two halvings; therefore $60=2T$ and $T=30$ days.
Inverse functions. If $(a,b)$ is on the graph of $f$, then $(b,a)$ is on the graph of $f^{-1}$: the table is obtained by swapping columns, and the graph is the reflection across the line $y=x$. For $f(x)=2^x$, the points $(0,1),(1,2),(2,4),(3,8)$ give $(1,0),(2,1),(4,2),(8,3)$, which correspond to $f^{-1}(x)=\log_2 x$.
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven from when it is turned on is given by the function $T(t) = 20 + 15t$, where $t$ is measured in minutes.…
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven from when 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 $200^\circ\text{C}$?
You can do it on paper and send a photo with your answer.
In a laboratory, the growth of a bacterial population is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \times 2^t$, where $t$…
In a laboratory, the growth of a bacterial population is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \times 2^t$, 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) Draw the graph of the function. What type of growth is observed?
You can do it on paper and send a photo with your answer.
The monthly profit $B(x)$, in pesos, 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 pesos, of an electronic components factory as a function of the number of units produced $x$ is modeled by the function $B(x) = -100x^2 + 6\,000x - 80\,000$.
- a) Find the break-even points (production for which the profit is zero).
- b) Generate a strategic table of values including the break-even points and the vertex.
- c) How many units must be produced to maximize the profit, and what is this maximum profit?
The mass $M(t)$ in grams of a radioactive isotope decreases over time $t$ (in days) according to the exponential function $M(t) = M_0 \times (0,5)^{\frac{t}{T}}$, where…
The mass $M(t)$ in grams of a radioactive isotope decreases over time $t$ (in days) according to the exponential function $M(t) = M_0 \times (0,5)^{\frac{t}{T}}$, where $M_0$ is the initial mass and $T$ is the half-life.
Given that at time $t=0$ there were $200\text{ g}$ of substance and after $60\text{ days}$ there are $50\text{ g}$ remaining:
- a) Calculate the values of the parameters $M_0$ and $T$.
- b) Construct a table of values for $t = 0, 30, 60, 90, 120$ and sketch the graph.
You can do it on paper and send a photo with your answer.
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.
You can do it on paper and send a photo with your answer.
It is desired to build a rectangular pen taking advantage of an existing wall for one of its sides. There are $20\text{ m}$ of wire fencing available for the other three…
It is desired to build a rectangular pen taking advantage of an existing wall for one of its sides. There are $20\text{ m}$ of wire 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 valid domain 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 maximise the area.
You can do it on paper and send a photo with your answer.
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 exercise book using symmetry with respect to the line $y=x$.
- c) Write the algebraic expression for $f^{-1}(x)$.
You can do it on paper and send a photo with your answer.
Reading a graph: domain, range, and asymptotes
You don't always build the graph: you also have to read it when it is already drawn.
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Domain: the values of $x$ over which there is a curve (the graph is projected onto the x-axis).
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Range: the values of $y$ it reaches (projected onto the y-axis).
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$f(a)$: you go up or down from $x=a$ to the curve and read the height.
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Asymptotes: lines the graph approaches without touching them.
This is how the cooling of a coffee looks: it starts at a high temperature and drops, approaching the room temperature without ever reaching it. The function is decreasing and the horizontal asymptote $y=T_{\text{room}}$ represents that equilibrium temperature.
To deduce a rational function $f(x)=\dfrac{ax+b}{cx+d}$ from its graph: the vertical asymptote gives the value that makes the denominator zero, the horizontal one is $y=\dfrac{a}{c}$, and the intercepts with the axes give the numerator. You set up these conditions and solve for the coefficients.
Observe the following graph of a function $f(x)$ defined by straight-line segments: From the graph, determine: a) The domain and range of the function. b) The value of…
Observe the following graph of a function $f(x)$ defined by straight-line segments:
From the graph, determine:
- a) The domain and 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 time $t$ (in minutes) since it is served. From…
The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of 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 approach as a large amount of time elapses? What physical meaning does the horizontal asymptote have?
- 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 axis intercepts):…
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 axis intercepts):
- a) Determine the domain and range of the function.
- b) Deduce the algebraic expression of the function $f(x)$.
Back to the beginning: the graph for the parking lot is a line whose y-intercept equals the flat fee and whose slope equals the cost per hour. Two calculated points are enough to draw it.