Using the Casio fx-9860 to Verify Graphs, Roots, and Numerical Values

Kia ora. In the previous lesson you corrected a calculus mini-set by separating method errors from calculation, justification, and communication gaps. This lesson adds a practical checking habit: use your Casio fx-9860 to catch mistakes early, while keeping the algebraic or calculus evidence that earns NCEA credit on your written paper.

Your calculator can quickly show a graph, approximate roots, intersections, function values, gradients at a point, and definite-integral values. Those are powerful checks. They are not a substitute for showing why a solution is correct. By the end of this lesson, you should be able to use the calculator to confirm your work, notice when something is wrong, and then fix the first incorrect written step.

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The exam-safe calculator principle

Treat the calculator as a verification tool, not as the source of your solution.

A strong written response has this order:

1. Choose and show the mathematical method.
2. Carry out the algebra or calculus clearly.
3. Use the calculator to check a root, graph feature, or numerical value.
4. If the check disagrees, find and correct the first line that caused the disagreement.
5. Write the final answer in the form the question requires, including exact values, units, intervals, or context.

The calculator may tell you that a root is approximately $2.618$, for example. But if the question can be solved exactly by factorising or the quadratic formula, your paper should still show that method.

Question asks you to...	Useful calculator check	Evidence still needed on paper
Solve an equation	Graph roots or graph intersections	Rearrangement, factorising, quadratic formula, logarithms, or another valid algebraic method
Find a point of intersection	Plot both functions and use ISCT	Set the functions equal and solve, then give both coordinates
Find a stationary point	Graph maximum or minimum	Differentiate, solve $f^{\prime }(x)=0$, and justify maximum or minimum
Find a gradient at a point	Numerical derivative	Differentiate to obtain $f^{\prime }(x)$, then substitute the given value
Evaluate a definite integral	Numerical integral	Find an anti-derivative and substitute the bounds
Check a substitution or arithmetic result	Direct numerical evaluation or Trace	Your original working and final interpretation

A useful rule is:

If a marker could not reconstruct your method from the written page, the calculator result alone is not enough.

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Set up a graph so that it can actually check your work

On the fx-9860, start in the GRAPH application. Enter expressions as $Y_1$, $Y_2$, and so on. Use the calculator’s dedicated $X$ variable key in graph mode rather than treating $x$ as a letter stored in memory.

The graph is only as reliable as its setup. Before trusting it, check:

• every coefficient and sign;
• brackets around any numerator or denominator;
• that your $x$-window includes the values you expect;
• that the $y$-window is large enough to show the relevant part of the graph.

For an unfamiliar polynomial, the standard window is a reasonable first look, but it is not guaranteed to show every turning point or root. Use V-Window to enter sensible limits, or use zoom tools to inspect the area that matters.

The Trace feature is also helpful for checking values. It moves a cursor along the graph and displays the corresponding $x$- and $y$-coordinates. This is useful for estimating whether your calculated value is plausible, but it is usually less accurate and slower than direct numerical evaluation.

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Checking roots: graph evidence versus algebraic proof

Suppose you solve:

$$x^2-5x+6=0$$

Your written method should be:

$$(x-2)(x-3)=0$$ $$x=2\text{or}x=3$$

Now enter $Y_1=X^2-5X+6$, draw the graph, and use G-Solv, then ROOT. The calculator should locate roots at $x=2$ and $x=3$.

This is an ideal check because it tests several things at once:

• Did you enter the original equation correctly?
• Did you factorise correctly?
• Did you lose a solution?
• Did you make a sign mistake?

However, the graph does not replace the factorisation. A graph may show an approximate root, look as if it touches the axis, or fail to reveal a root that lies outside the current window.

A root check is especially valuable after the quadratic formula

For a less factorisable equation such as

$$2x^2+x-1=0$$

you might write:

$$x=\frac{-1\pm \sqrt{1^2-4(2)(-1)}}{2(2)}$$ $$x=\frac{-1\pm 3}{4}$$ $$x=\frac{1}{2}\text{or}x=-1$$

Graphing $Y_1=2X^2+X-1$ should show roots at $-1$ and $0.5$. If it does not, do not just replace your answer with the calculator result. Check, in order:

1. Did you enter $a$, $b$, and $c$ correctly?
2. Did you include brackets in the numerator of the quadratic formula?
3. Did you calculate the discriminant correctly?
4. Did you divide both parts of the numerator by $2a$?

This preserves the useful part of calculator checking: it tells you that an error exists, while your written reasoning tells you where it occurred.

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Checking intersections by graphing both sides

An equation can be checked as an intersection. For example:

$$x^3+4x^2+2x-2=-0.5x$$

Enter the left side as $Y_1$ and the right side as $Y_2$. Then draw both graphs and select G-Solv, then ISCT for intersection. The coordinate shown is a numerical approximation.

The calculator image shows why an intersection is more informative than a sketch alone: it identifies an approximate coordinate. But the decimal display also reveals a limitation. A graphing calculator normally gives a numerical approximation, so it may not preserve an exact surd, fraction, or logarithmic form.

For an algebra question, your written structure is still:

$$x^3+4x^2+2x-2=-0.5x$$ $$x^3+4x^2+2.5x-2=0$$

Then continue with the appropriate algebraic method. Use ISCT afterward to check that the solution lies at the displayed intersection.

When a question has two functions, label them carefully on paper if needed:

$$f(x)=x^3+4x^2+2x-2$$ $$g(x)=-0.5x$$

Then an intersection satisfies $f(x)=g(x)$. This makes the logic visible to the marker.

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Calculator checks for calculus: numerical is not symbolic

Your fx-9860 can evaluate the derivative at one particular value of $x$. It does not give the full derivative expression in a form that replaces differentiation.

Suppose:

$$f(x)=x^3-3x$$

Differentiate by hand:

$$f^{\prime }(x)=3x^2-3$$

At $x=2$,

$$f^{\prime }(2)=3(2{)}^2-3=9$$

Use the calculator’s numerical differentiation feature to check that the gradient of $x^3-3x$ at $x=2$ is approximately $9$. This confirms the final numerical evaluation, but the calculator display does not show the power rule that produced $3x^2-3$.

Stationary points: use the graph to check, not classify

For the same function,

$$f(x)=x^3-3x$$

the stationary points come from:

$$f^{\prime }(x)=0$$ $$3x^2-3=0$$ $$x^2=1$$ $$x=-1\text{or}x=1$$

The calculator’s G-Solv maximum and minimum tools should show turning points near $(-1,2)$ and $(1,-2)$. This is a useful confirmation that you found both stationary values.

But your paper still needs a reason that classifies them. For example, using the sign of $f^{\prime }(x)$:

Interval	Sign of $f^{\prime }(x)$	Behaviour of $f(x)$
$x<-1$	Positive	Increasing
$-1<x<1$	Negative	Decreasing
$x>1$	Positive	Increasing

Therefore, the curve changes from increasing to decreasing at $x=-1$, so it has a local maximum there. It changes from decreasing to increasing at $x=1$, so it has a local minimum there.

The graph supports this conclusion visually. The derivative-sign reasoning justifies it.

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Checking numerical values and definite integrals

Direct calculator evaluation is excellent for checking arithmetic after substitution. If you need to evaluate a polynomial at $x=38$, entering the expression with brackets can quickly catch arithmetic slips.

For example, if your written work gives:

$$F(38)=664240$$

enter the original function and $38$ into the calculator. If the displayed number differs, review the substitution line by line. In particular, check negative signs, powers, and brackets.

For a definite integral, write the calculus first. Consider:

$${\int }_2^4{x}^{2}dx$$

Your written method is:

$${\int }_2^4{x}^{2}dx={\left[\frac{x^3}{3}\right]}_2^4$$ $$=\frac{4^3}{3}-\frac{2^3}{3}$$ $$=\frac{56}{3}$$ $$=18.666\dots$$

A calculator’s numerical integration feature should return approximately $18.666\dots$. This checks your answer, but it will generally show a decimal rather than the exact value $\frac{56}{3}$.

For Level 2 calculus, keep the exact form unless the question asks for a decimal approximation or the context requires one. The calculator is particularly useful here for detecting a missed lower bound, such as accidentally calculating only $\frac{4^3}{3}$.

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A fast checking routine for timed papers

When working under time pressure, do not graph every question. Use the calculator at points where it can prevent a costly chain of errors.

Use a graph check when you have:

• solved an equation with an unexpected decimal or a difficult factorisation;
• found roots of a polynomial;
• solved an equation by setting two functions equal;
• found stationary values and want to check whether the graph shape makes sense;
• obtained a suspicious maximum, minimum, or intersection.

Use a numerical check when you have:

• substituted into a long polynomial;
• evaluated a derivative at a particular time or position;
• calculated a value at stationary points and endpoints;
• evaluated a definite integral after finding an anti-derivative.

Do not rely on a calculator check when the key issue is:

• selecting the correct method;
• writing a proof or explanation;
• deciding what a derivative means in context;
• showing why a stationary point is a maximum or minimum;
• giving an exact algebraic form when the display is decimal.

A compact note in your working can help you use checks without making them the argument:

Calculator check: roots agree with $x=-1$ and $x=\frac{1}{2}$.

That note is optional; the important evidence remains the algebra above it.

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When the calculator disagrees with you

A disagreement is useful feedback, not a reason to panic or overwrite your work. Use this diagnostic sequence:

1. Check the calculator entry. A missing bracket or incorrect negative sign can create a false disagreement.
2. Check the viewing window. A root or turning point outside the visible interval will not appear.
3. Check your first mathematical transformation. For an equation, this might be moving a term across the equals sign. For calculus, it might be the derivative.
4. Check substitution separately. Recalculate powers and bracketed values before combining terms.
5. Keep the correct written method. Replace only the step that is actually wrong.

For instance, if you differentiated

$$6x^2-4x+3$$

as

$$12x-4x$$

a numerical gradient check may reveal an error. The first incorrect line is the derivative: the derivative of $-4x$ is $-4$, not $-4x$. Correct it to:

$$\frac{d}{dx}(6x^2-4x+3)=12x-4$$

This is much better than simply copying the calculator’s numerical gradient, because you have repaired the underlying rule.

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Key takeaways

Your Casio fx-9860 is most useful when it helps you verify work you have already set up mathematically:

• Use GRAPH, Trace, and G-Solv to check graph shape, roots, turning points, and intersections.
• Treat graph and calculator answers as numerical approximations unless your written algebra establishes an exact value.
• Use numerical differentiation to check a gradient at a point, not to replace finding $f^{\prime }(x)$.
• Use numerical integration to check a definite-integral value, not to replace anti-differentiation and substitution of bounds.
• If a check disagrees with your answer, locate and fix the first incorrect step rather than replacing your working.

You have now completed the one-week mock sprint: interpreting marking evidence, selecting methods, completing timed mini-sets, correcting them precisely, and using the calculator responsibly. The next module strengthens the algebraic manipulation that underpins AS91261, beginning with index laws for negative and fractional exponents.