From a Formula to a Sketch
A formula gives a rule for calculating outputs from inputs. A graph gives a visual picture of those input-output pairs: each pair \((x,f(x))\) is plotted as a point in the coordinate plane. When the formula is known, a value table is a practical way to calculate a set of points before sketching the larger graph.
A table does not, by itself, tell you how to join its points. The formula and its domain matter. A straight line formula supports a straight-line graph; a quadratic formula supports a curved graph; and a rule with an excluded input cannot be drawn as an unbroken curve through that input. The aim is to use the table to guide a sketch of the formula, not to connect dots automatically.
Build and Plot a Value Table
Choose inputs that help reveal the graph. For an introductory sketch, convenient values near zero are often a good start. For a quadratic, inputs on both sides of a central value can show the curve’s shape. For a formula with a restricted input, include values on both sides of the restriction when useful, but never use the excluded input itself.
For each chosen input, substitute into the formula and calculate the output. Record the input and output in matching columns. The ordered pair is always written with the input first and the output second: \((x,f(x))\). On the graph, move horizontally to \(x\), then vertically to \(f(x)\).
Before plotting, choose axis scales that show all the values clearly. The scale on the horizontal axis need not match the scale on the vertical axis, but the tick marks on each axis must be evenly spaced and labelled consistently. If some outputs are negative, make sure the vertical scale includes them. A plotted point’s position should reflect its numerical coordinates, not just its row in the table.
Select values that are allowed by the formula and reveal relevant parts of its behavior.
Substitute each input into the formula and check the arithmetic.
Place each point at horizontal coordinate \(x\) and vertical coordinate \(f(x)\).
Connect points only when the formula supports that connection, and respect any gaps or jumps.
Worked Example: Sketching a Linear Formula
Worked Example: Sketching a Linear Formula
Sketch \(f(x)=2x-1\) using inputs from \(-2\) to \(2\). Substitute each input into the formula:
| \(x\) | \(f(x)=2x-1\) | Point to plot |
|---|---|---|
| \(-2\) | \(-5\) | \((-2,-5)\) |
| \(-1\) | \(-3\) | \((-1,-3)\) |
| \(0\) | \(-1\) | \((0,-1)\) |
| \(1\) | \(1\) | \((1,1)\) |
| \(2\) | \(3\) | \((2,3)\) |
Plot the five points and draw the straight line through them. A linear formula defines a line, so joining these points with a straight line is supported by the model. Check the points by substituting their inputs back into \(2x-1\): the five outputs are \(-5,-3,-1,1,3\), exactly the table values.
Worked Example: Sketching a Quadratic Formula
Worked Example: Sketching a Quadratic Formula
Sketch \(g(x)=x^2-4x+3\). Use several inputs around \(x=2\), where the formula can also be written in vertex form:
Calculate the table values directly from the original formula:
| \(x\) | \(g(x)\) | Point to plot |
|---|---|---|
| \(0\) | \(3\) | \((0,3)\) |
| \(1\) | \(0\) | \((1,0)\) |
| \(2\) | \(-1\) | \((2,-1)\) |
| \(3\) | \(0\) | \((3,0)\) |
| \(4\) | \(3\) | \((4,3)\) |
The equivalent form \((x-2)^2-1\) shows that the graph has its lowest point at \((2,-1)\) and is symmetric about the vertical line \(x=2\). Plot the points, then sketch a smooth U-shaped curve through them, with its bottom at \((2,-1)\). Check by substituting \(x=2\) into the original formula: \(2^2-4(2)+3=4-8+3=-1\), as recorded. The table guides the curve, but the formula determines its shape between the listed inputs.
Why a Table Alone Does Not Determine the Graph
A finite table gives only a finite number of points. Many different formulas can pass through the same points and have different values between them. For instance, suppose a table lists three distinct inputs \(a,b,c\) and their outputs from a formula \(f\). Define another formula by
At \(x=a\), the added product is zero because it contains the factor \(a-a\). The same is true at \(x=b\) and \(x=c\). Thus \(h\) and \(f\) agree at all three listed inputs. At other inputs, the added product may not be zero, so the formulas may give different outputs. This is why points from a value table cannot, by themselves, establish the exact curve between those points. The given formula supplies the additional information needed for a reasonable sketch.
Why a Linear Formula Gives a Straight Line
For a linear formula, two distinct inputs determine the line, and every other value lies on that same line. Here is a direct justification. Let \(f(x)=mx+b\), and choose two distinct inputs \(x_1\) and \(x_2\). For any number \(t\) between \(0\) and \(1\), consider the input between them given by \((1-t)x_1+tx_2\). Substitution gives:
The input is a weighted position between \(x_1\) and \(x_2\), and its output is the same weighted position between \(f(x_1)\) and \(f(x_2)\). Therefore, points between the two selected points lie on the straight segment joining them. This explains why connecting points with a straight line is appropriate for a linear formula. It does not justify connecting points with straight segments for every formula.
Worked Example: A Restricted Input and a Gap
Worked Example: A Restricted Input and a Gap
Sketch \(q(x)=\dfrac{1}{x-1}\). Its denominator is zero at \(x=1\), so \(x=1\) is not an allowed input. Calculate values on each side:
Plot the corresponding points in two groups: those with \(x<1\) and those with \(x>1\). Sketch a curve on each side, but leave a gap at \(x=1\). The formula is undefined there, so there is no point on the graph with input \(1\). Check the restriction directly: substituting \(1\) into the denominator gives \(1-1=0\), and division by zero is not defined.
When Should Points Be Connected?
The correct connection depends on the rule, not on a general instruction to join neighboring rows of a table. Use the formula to decide what the sketch should show.
| Formula or rule | Sketching guidance |
|---|---|
| Linear formula, such as \(2x-1\) | Plot points and connect them with a straight line. |
| Quadratic formula, such as \(x^2-4x+3\) | Plot points and sketch the corresponding curved graph. |
| Formula with an excluded input | Do not draw the graph through the excluded input; show the separate allowed parts. |
| Step rule | Respect the jumps and endpoint conventions instead of drawing an unbroken curve. |
For a formula defined only at certain inputs, the graph contains points only for those inputs. In an applied model, the context may further restrict which inputs are meaningful, even if the formula can be calculated elsewhere. The domain therefore affects where the graph can be drawn.
A Common Mistake: Connecting Every Point
Suppose someone calculates values for \(q(x)=\dfrac{1}{x-1}\) on either side of \(1\), then draws a line from the point \((0.9,-10)\) to \((1.1,10)\). This incorrectly suggests that the graph passes through \((1,0)\), because that point lies halfway along the drawn segment. But the formula does not define \(q(1)\): its denominator would be \(1-1=0\). The correction is to leave a gap at \(x=1\) and sketch the two sides separately. More generally, do not join table entries just because they appear in consecutive rows; first check the formula’s domain and shape.
Check Your Understanding
Use the formula, domain, and plotted values together when answering.
- For \(r(x)=3x+2\), calculate \(r(-1)\), \(r(0)\), and \(r(2)\), and list the three points to plot.
- For \(s(x)=(x-1)^2\), calculate its values at \(x=0,1,2\). What do those points suggest about the shape?
- Why is a value table with only three inputs not enough to determine every point on a graph?
- For \(u(x)=\dfrac{1}{x+2}\), which input must be excluded, and why should the sketch not pass through it?
- What should you check before deciding to connect two points from a value table?
