How to Graph a Function

A function assigns one output to each allowed input. The graph is the set of pairs $(x,f(x))$. This lesson graphs two rules from scratch: a straight line, then a curve that turns. Each step shows the arithmetic, the point you plot, and the line you type.

The handbook page on pairs $(x,f(x))$ on a live board names domain, range, and graph, then goes on to increase, decrease, and the diagonal flip. Stay here until both figures below match the tables.

Graph a line

The first rule is $y=2x+1$.

Write the line

The graph starts as one line of text:

y = 2x + 1

2x means two times $x$. 2*x is the same product. Spaces do not change the rule, so y=2x+1 is the same line. To choose the name yourself, type f(x) = 2x + 1. That is the same curve, and the name is the $f$ you wrote.

A bare 2x + 1 also draws, and the row comes out as $f(x)=2x+1$. In this lesson every graph is typed with y =, so the algebra row reads f: y = 2x + 1.

That line is the same shape as a Desmos expression row. Sliders and the Desmos rows themselves are on how to use Desmos.

The intercept, then the slope

Put $x=0$ into the rule.

$$2\cdot 0+1=1.$$

Plot $(0,1)$. This point sits on the vertical axis. It is the intercept.

Put $x=1$ in next.

$$2\cdot 1+1=3.$$

Plot $(1,3)$. From $(0,1)$ you moved right $1$ and up $2$. That rise over that run is the slope, $2$. Two points already determine the line. The rest of the table checks that the same step keeps landing on it.

Three more rows

Keep the multiplication visible. Do not jump to the finished height.

$$\begin{array}{rlrlr}x=2& :& 2\cdot 2+1& =5,& \text{point }(2,5).\\ x=3& :& 2\cdot 3+1& =7,& \text{point }(3,7).\\ x=4& :& 2\cdot 4+1& =9,& \text{point }(4,9).\end{array}$$

Now the table is only a record of those five calculations.

Plot $(0,1)$ and $(1,3)$ first. Then plot $(2,5)$, $(3,7)$, and $(4,9)$. All five sit on the line through the first pair. Draw that line. On the board above, those five marks are fixed. They are the table, and they do not move.

Check a point that was not in the table

The line claims every input, not only the five you listed.

$$x=1.5:2\cdot 1.5+1=4.$$

$(1.5,4)$ sits halfway in height between $(1,3)$ and $(2,5)$, and halfway in input as well.

$$x=2.5:2\cdot 2.5+1=6.$$

$(2.5,6)$ sits halfway between $(2,5)$ and $(3,7)$. If either check misses the line, a row of the table was added wrong.

Type it, then add the points

Open the graphing workbench. No account is required. In Algebra, type

y = 2x + 1

and press Enter. The algebra row reads f: y = 2x + 1. The board shows the line. You do not plot the five rows before the line exists. The expression is the whole set of pairs. Zooming changes the window, not the rule.

On the next row type

A = (0, 1)

Press Enter. $A$ is the intercept, and it sits on the line. Then type

B = (2, 5)

$B$ sits on the line too, because the table already gave height $5$ at $x=2$. Then type

C = (2, 4)

$C$ is not on the line. At $x=2$ the rule requires height $5$, so a mark at height $4$ hangs below the graph. A point you type as a fixed pair does not follow the rule. It only lies on the graph when its two numbers match a row.

Drag the line

On the board at the top of this lesson, $P$ starts at $(2,5)$. Drag it. It has to stay on $y=2x+1$, so the height you read is the value of the formula. Move right by $1$ and the height rises by $2$. The five fixed marks stay where the table put them.

Graph a curve that turns

Two points determined the line. They do not determine the next rule. $y=x^2-2x-3$ falls, reaches a low point, and rises. A straight stroke through two of its points misses the rest.

Write the square with a caret

Type

y = x^2 - 2x - 3

The square is x^2, with a caret. x2 is not that square. The row may be accepted, and still no curve appears, because that name has no height. Parentheses give the same curve under another writing:

f(x) = (x-1)^2 - 4

$(x-1{)}^2$ is the square of the distance from $1$. Subtract $4$ and the low point is $4$ below the axis, at input $1$. Either line is the graph. This lesson types the first one.

Five rows, with the square written out

$$\begin{array}{rlrlr}x=-1& :& (-1{)}^2-2(-1)-3& =1+2-3=0,& \text{point }(-1,0).\\ x=0& :& 0-0-3& =-3,& \text{point }(0,-3).\\ x=1& :& 1-2-3& =-4,& \text{point }(1,-4).\\ x=2& :& 4-4-3& =-3,& \text{point }(2,-3).\\ x=3& :& 9-6-3& =0,& \text{point }(3,0).\end{array}$$

Read the heights in order: $0,-3,-4,-3,0$. They fall, then rise. The low point in the table is $(1,-4)$. The heights match on either side of $x=1$: $x=0$ and $x=2$ are both $-3$, and $x=-1$ and $x=3$ are both $0$. Those last two points sit on the horizontal axis. They are the cuts.

The vertex formula and the test that counts the cuts are on the handbook page for quadratic functions. Here the job is the picture: plot the five points, then draw a smooth curve through them. Do not join them with straight segments. The segment from $(-1,0)$ to $(1,-4)$ passes through $(0,-2)$. The table point $(0,-3)$ is below that segment, so a straight stroke misses the curve.

One input past the table

The right arm has only been checked up to $x=3$.

$$x=4:4^2-2\cdot 4-3=16-8-3=5.$$

$(4,5)$ is above the axis, to the right of $(3,0)$. The curve keeps rising there. The same step on the left, $x=-2$, gives $4+4-3=5$, so $(-2,5)$ matches it. The symmetry you saw in the table continues past the five rows.

Type the curve, then the three marks

In Algebra on the graphing workbench, type

y = x^2 - 2x - 3

and press Enter. The row reads f: y = x^2 - 2x - 3, and the board shows the curve turning at a low point. Then type the three marks from the table, one row at a time:

V = (1, -4)

R = (-1, 0)

S = (3, 0)

$V$ sits at the bottom of the curve. $R$ and $S$ sit where the curve meets the horizontal axis. If a mark floats off the curve, the pair does not match the rule you just typed.

Drag the curve

On the board above, $P$ starts at the low point $(1,-4)$. Drag it. It stays on $y=x^2-2x-3$.

Move $P$ to the left of $x=1$. As the input gets smaller, the height rises: from $(1,-4)$ toward $(0,-3)$ and then $(-1,0)$. Move $P$ to the right of $x=1$ and the height rises the other way, through $(2,-3)$ toward $(3,0)$. The three fixed marks stay put. They are the cuts and the low point, not passengers on $P$.

Say it in the chat

The Algebra rows above are what you type when you already know the line. The other way is a sentence. Open chat with a live board. You can try it without an account.

The message box sits at the bottom of the chat. Its hint is “Ask anything…”. Write the rule in that box and press Send. The graph is the pane under the chat on a narrow window, and the pane beside the chat on a wide one. That graph is where the sentence is drawn. You do not also type the expression into Algebra for this path.

The line, as one message

Send this:

Graph $y=2x+1$. Mark $(0,1)$ and $(2,5)$ on the line, and put a point on the line I can drag.

The graph under the chat draws the line, those two marks, and a point that stays on the line when you drag it.

Move that point right by $1$ and its height rises by $2$. To put the rest of the table on the same graph, send a second message in the same chat:

Also mark $(1,3)$, $(3,7)$, and $(4,9)$.

Those three join the first two. The line is unchanged, because each pair is a row of $y=2x+1$.

One more message checks a point that misses:

Also mark $(2,4)$.

At $x=2$ the rule requires height $5$, so $(2,4)$ hangs below the line. The line does not bend down to meet it.

The curve, as one message

Press New chat, so the line is not still on the graph. Then send:

Graph $y=x^2-2x-3$. Mark the low point $(1,-4)$ and the axis cuts $(-1,0)$ and $(3,0)$, and put a point on the curve I can drag.

The graph under the chat draws the turn, those three marks, and a point that stays on the curve. Drag it left of $x=1$: as the input gets smaller, the height rises. Drag it right of $x=1$: the height rises the other way. The three marks stay put.

If the message said $x2$ and the curve did not appear, send the correction in that same chat:

Use $x^2$ with a caret, not $x2$, and draw the curve again.

The square is the caret. The graph follows that correction.

The same steps on the next rule

Write the rule as y = …. Build a short table that includes an intercept, and a turn when the rule has one. Plot those points. Compute one input that was not in the table. Then either type that y = … line into Algebra on the graphing workbench, or send it as a sentence in the message box and read the graph under the chat.

GoodMaith is the place those lines are objects on a board. The message box asks for the figure. The graph under the chat is where it shows up.