CG4 - Sketching and plotting polynomial graphs

A graph turns an equation into a picture of all its solutions. You will plot straight lines and polynomial curves accurately, and make quick sketches from roots, intercepts and signs. You will also learn when a table of values can hide a bend, so that a neat drawing does not tell the wrong story.

Plotting straight lines

Could three correct plotted points persuade you to draw the wrong graph? For $y=x(x-1)(x+1)$ , the values at $x=-1, 0, 1$ are all zero. Joining just those points along the axis would conceal the curve completely: at $x=\frac12$ , its height is $-\frac38$ . A straight line and a polynomial curve need different plotting decisions.

You already know

Substitute a chosen $x$ into an expression to find $y$ . The solutions of $\mathrm{f}(x)=0$ are its roots. A factor $x-a$ gives a root $a$ , by the factor theorem.

Start with lines, where the plotting job is simpler. Two distinct points fix a straight line; a third point checks the arithmetic. Choose a clear scale on each axis, label both axes and plot coordinate pairs in the order $(x, y)$ . The two axis scales may differ, but each scale must be uniform.

Plot and sketch

To plot a graph, calculate and mark accurate points using a stated scale, then draw the appropriate line or smooth curve. To sketch a graph, show its main shape and features, including labelled intercepts; exact scale is unnecessary. An intercept is where a graph meets an axis: use $x=0$ for the vertical intercept and $y=0$ for horizontal intercepts.

A linear function has the form $y=mx+c$ , with constants $m$ and $c$ ; its graph is a straight line. This form is not on the formulae sheet: learn it. We will interpret the gradient in more detail in the next lesson.

Worked example 1: plotting a rising line

Question. Plot $y=2x-3$ for $-1\le x\le3$ , stating both axis intercepts.

Plan. Calculate three points, draw the straight line through them, then set each coordinate to zero in turn.

\begin{array}{c|rrr} x&-1&1&3\\\hline y&-5&-1&3 \end{array}

\begin{aligned} y\big|_{x=0}&=-3 && \text{vertical intercept}\\ 0&=2x-3 &\Longrightarrow x&=\frac32 && \text{horizontal intercept} \end{aligned}

Use one-unit scales, mark $(-1, -5)$ , $(1, -1)$ and $(3, 3)$ , and draw one straight segment across the specified interval. The left panel below shows the result.

Why use three points? The third checks that a substitution or plotting error has not bent a line that should be straight.

Answer. A straight line through the plotted points, with intercepts $(0, -3)$ and $(\frac32, 0)$ .

Check. The vertical intercept also lies on the line through the table points, and $2(\frac32)-3=0$ .

The next example changes only the sign of the coefficient of $x$ . The vertical intercept stays put while the line falls instead of rising.

Worked example 2: plotting a falling line

Question. Plot $y=-2x-3$ for $-1\le x\le3$ , stating both axis intercepts on the full line.

Plan. Recalculate the same three heights and find the intercepts by substitution.

\begin{array}{c|rrr} x&-1&1&3\\\hline y&-1&-5&-9 \end{array}

\begin{aligned} y\big|_{x=0}&=-3 && \text{same vertical intercept}\\ 0&=-2x-3 &\Longrightarrow x&=-\frac32 && \text{horizontal intercept} \end{aligned}

Mark $(-1, -1)$ , $(1, -5)$ , $(3, -9)$ and draw their straight segment. The horizontal intercept belongs to the extended line, outside the requested interval; the right panel shows that extension dashed.

Why mention the interval? A correct feature of the full line need not lie on the portion you were asked to plot.

Answer. The line falls through the table points; the full line's intercepts are $(0, -3)$ and $(-\frac32, 0)$ .

Check. $-2(-\frac32)-3=0$ , and each increase of $2$ in $x$ lowers $y$ by $4$ .

The plots make the sign change visible while keeping the scales fixed.

[DIAGRAM: asset_name: Two accurately plotted straight lines; asset_slug: cg20-linear-plots; caption: Changing the sign of the coefficient changes rising to falling while keeping the vertical intercept; description: Two coordinate grids plot y equals 2x minus 3 and y equals minus 2x minus 3. Clay points mark the calculated values at x equals minus 1, 1 and 3. Blue solid segments show the requested interval minus 1 to 3; dashed extensions reveal the full lines and their axis intercepts.]
Changing the sign of the coefficient changes rising to falling while keeping the vertical intercept

A line is not a collection of isolated plotted dots: it includes every point between them that satisfies the equation.

Watch out

Wrong: plot $(3, -1)$ when the table says $x=-1$ , $y=3$ . Coordinate order matters. Move horizontally to $x$ , then vertically to $y$ , and check one substitution after drawing.

Now keep the falling coefficient and change only the constant.

Sketching polynomial graphs from factors

Factors reveal where a polynomial meets the horizontal axis and whether its sign changes there; its highest-power term tells you how the graph behaves far to the left and right.

A polynomial is a sum of constant multiples of non-negative whole-number powers of $x$ , such as $x^3-2x+1$ . Expressions such as $1/x$ and $\sqrt x$ are not polynomials. Its degree is its highest power with a non-zero coefficient. Polynomial graphs are continuous and smooth: draw no jumps, gaps or sharp corners.

Multiplicity of a root

A root $a$ has multiplicity $k$ if $(x-a)^k$ is a factor and the remaining factor is non-zero at $a$ . For an odd multiplicity the graph crosses the axis; for an even multiplicity it touches the axis and turns back. For example, $(x-1)^2$ has a double root at $1$ ; two equal factors give one intercept, not two different intercepts.

Here is why the rule works. Near $a$ , the remaining factor keeps the same sign. An odd power of $x-a$ changes sign as $x$ passes $a$ ; an even power stays non-negative. A triple root crosses with a flattened shape, whereas a double root touches. These multiplicity and sign rules are not on the formulae sheet: learn them.

For end behaviour, the highest power dominates when $|x|$ is large. With a positive leading coefficient, an even degree rises at both ends, while an odd degree falls to the left and rises to the right. A negative leading coefficient reverses both directions. These end-behaviour rules are not on the formulae sheet: learn them.

Worked example 3: three distinct roots

Question. Sketch $y=(x+2)(x-1)(x-2)$ , showing the axis intercepts.

Plan. Find roots and the vertical intercept, check the sign between roots, then join smoothly with the correct ends.

\begin{aligned} y=0&\Longrightarrow x=-2, 1, 2 && \text{set each factor to zero}\\ y\big|_{x=0}&=2(-1)(-2)=4 && \text{vertical intercept}\\ (x+2)(x-1)(x-2)&=x^3+\cdots && \text{positive cubic leading term} \end{aligned}

\begin{array}{c|rrrr} x\text{ interval}&x<-2&-2<x<1&1<x<2&x>2\\\hline \text{sign of }y&-&+&-&+ \end{array}

Draw the curve from the lower left, crossing at $(-2, 0)$ , passing through $(0, 4)$ , crossing at $(1, 0)$ and $(2, 0)$ , then rising to the right. The left panel below shows these features. Exact turning-point coordinates are not needed for this sketch.

Why check signs? Roots locate crossings, but signs tell you which side of the axis each joining section must occupy.

Answer. A smooth cubic through the four labelled intercepts, crossing at every root.

Check. At $x=\frac32$ , $y=\frac72\times\frac12\times(-\frac12)=-\frac78<0$ , agreeing with the dip between $1$ and $2$ .

Change just the final factor from $x-2$ to $x-1$ . Two simple roots merge into one double root, so a crossing becomes a touch.

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CG4 - Sketching and plotting polynomial graphs

Plotting straight lines

Sketching polynomial graphs from factors

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