IB Math Command Terms

Explain – Give a detailed account including reasons or causes. For example, if asked to explain why the graph of a sine function is periodic, you would describe how the sine function, $y = \sin(x)$, repeats its values in regular intervals. You would explain that this periodic behavior arises from the function's definition as the y-coordinate of a point on the unit circle as it completes one full revolution. The period of the sine function is $2\pi$2π radians, meaning the function’s pattern repeats every $2\pi$2π units along the x-axis.

Find – Obtain an answer showing relevant stages in the working. For instance, if asked to find the value of a function at a specific point, such as $f(x) = 3x^2 + 2x - 5$f(x)=3x$2$+2x−5 when $x = 4$x=4, you would first substitute $4$4 into the function. This involves calculating $3(4)^2 + 2(4) - 5$3(4)$2$+2(4)−5. Then, simplify the expression step by step: first, calculate $4^2 = 16$$2$, then $3 \times 16 = 48$3×16=48, add $2 \times 4 = 8$2×4=8, and finally subtract $5$5. This process will give you the final answer. 

Hence –  Use the preceding work to obtain the required result. For example, if you have previously calculated the derivative of a function $f(x) = 2x^3 + 5x$f(x)=2x$3$+5x and found it to be $f'(x) = 6x^2 + 5$f′(x)=6x2+5, and you are then asked to find the derivative at $x = 3$x=3, you would use the result from the previous step. Substitute $x = 3$x=3 into $f'(x)$f′(x), which involves calculating $6(3)^2 + 5$6(3)$2$+5. This process will lead you to the final value of the derivative at that specific point.

Hence or otherwise – It is suggested that the preceding work is used, but other methods could also receive credit. For instance, if you have previously determined that the integral of a function $f(x)$ from $a$ to $b$ is approximated using the trapezoidal rule, and you are then asked to estimate this integral, you would use the trapezoidal rule result from the previous calculation. Substitute the values into the trapezoidal formula to estimate the integral's value. Alternatively, you could use another numerical integration method to achieve a similar estimate.

Identify –   Provide an answer from a number of possibilities. For example, if you are given a list of possible values for $x$x in the equation $x^2 - 9 = 0$x$2$−9=0 and asked to identify the correct values that satisfy the equation, you would test each possibility. In this case, you identify the correct solutions by substituting each potential value into the equation. For $x = 3$ and $x = -3$x=−3, you find that both satisfy the equation because substituting them into $x^2 - 9$$2$ yields zero. Therefore, $x = 3$ and $x = -3$x=−3 are identified as the solutions.

Integrate – Obtain the integral of a function. For example, if you are asked to integrate the function that is 2x with respect to x, you would apply the basic rule for integrating polynomials. For the function 2x, integrating it will give you x² plus a constant.

Interpret –  Use knowledge and understanding to recognize trends and draw conclusions from given information. For example, if you are given a set of data points for a function and a corresponding graph showing the trend, you might interpret the graph to understand how the function behaves. If the graph shows that as $x$x increases, $y$y initially increases rapidly but then levels off, you could interpret this to mean that the function approaches a horizontal asymptote. This understanding would help you conclude that the function has a limit as $x$x approaches infinity, based on the behavior of the graph.

Investigate –  Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions. For example, if you are asked to investigate the relationship between the variables $x$ and $y$ in a given function, you might start by observing how changes in $x$x affect $y$y by plotting the function on a graph. You could then systematically examine different values of $x$x to see how $y$y responds, looking for patterns or trends. This might involve calculating specific points, examining intervals where the function increases or decreases, and testing hypotheses about the function's behavior. Your investigation would aim to establish facts about the function’s characteristics and reach new conclusions about how $x$x and $y$ are related.

Justify –  Give valid reasons or evidence to support an answer or conclusion. For example, if you are asked to justify why a particular function has a maximum value at a certain point, you would start by showing that you have found the critical points of the function by setting its derivative to zero. You would then use the second derivative test to confirm that the point is a maximum. To support your conclusion, you would provide evidence from these calculations, such as demonstrating that the second derivative is negative at that point.

Label – Add labels to a diagram. For instance, if you are given a diagram of a triangle and asked to label it, you would start by identifying key components such as the vertices, sides, and angles. You would then add appropriate labels to the diagram, such as naming the vertices A, B, and C.

List – Give a sequence of brief answers with no explanation. If you're asked to list the properties of quadratic functions, you would simply provide: vertex, axis of symmetry, parabola direction, y-intercept, and x-intercept.

Plot – Mark the position of points on a diagram. For instance, if asked to plot the coordinates, you would mark their positions on the graph.

Predict – Give an expected result. For example, if you are instructed to predict the future value of a variable, you would use a formula to extrapolate from the existing data.

Prove – Use a sequence of logical steps to obtain the required result in a formal way. For example, if you are asked to prove that the sum of the interior angles of a triangle is 180 degrees, you would use a sequence of logical steps. First, you might draw a triangle and extend one of its sides to form a straight line. Then, you would show that the angles on a straight line add up to 180 degrees and use this to demonstrate that the sum of the interior angles of the triangle is also 180 degrees.

Show – Give the steps in a calculation or derivation. For example, if you are asked to show how to calculate the area of a triangle, you would start by using the formula for the area, which is half the product of the base and the height. If the base of the triangle is 4 units and the height is 5 units, you would first multiply the base by the height to get 20. Then, you would multiply 20 by 0.5 to find that the area is 10 square units.

Show that – Obtain the required result (possibly using information given) without the formality of proof. “Show that” questions do not generally require the use of a calculator. For instance, if you are asked to show that the quadratic equation $x^2 - 4x + 4 = 0$x$2$−4x+4=0 can be factored into $(x - 2)^2$(x−2)$2$, you would start by expanding $(x - 2)^2$(x−2)$2$ to verify that it equals $x^2 - 4x + 4$$2$. Thus, you show that $(x - 2)^2$(x−2)$2$ is indeed the correct factorization of the quadratic expression. This involves substituting the factorized form into the original equation and confirming that both sides are equivalent.

Sketch – Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features. For example, if asked to sketch the graph of the function $y = x^2 - 4$y=x$2$−4, you would draw a parabolic curve that opens upwards. The sketch should include the vertex of the parabola at $(0, -4)$(0,−4), and the curve should intersect the x-axis at $x = \pm 2$x=±2. Your diagram should be labelled with these key features to clearly illustrate the general shape and key points of the function. The difference between "sketch" and "draw" is that a "sketch" does not require precise tools like a ruler and does not need to be as exact. A sketch is intended to give a general idea of the shape or relationship, highlighting key features without a focus on exact measurements.

Solve – Obtain the answer(s) using algebraic and/or numerical and/or graphical methods. For example, if you need to find the roots of the equation $x^2 - 4x - 5 = 0$$2$, you can solve it by factoring the equation to $(x - 5)(x + 1) = 0$(x−5)(x+1)=0 and setting each factor to zero.

State – Give a specific name, value or other brief answer without explanation or calculation. For example, if asked to state the maximum value of the quadratic function $f(x) = -x^2 + 4x + 1$f(x)=−x$2$+4x+1, you would simply respond with "4" without further explanation or calculation.

Suggest – Propose a solution, hypothesis or other possible answer. For instance, if asked to suggest a method to determine if a quadratic function opens upwards or downwards, you might propose examining the sign of the coefficient of the squared term. You would suggest that if the coefficient is positive, the function opens upwards; if it is negative, the function opens downwards.

Verify –  Provide evidence that validates the result. For example, if you are asked to verify that a solution to a quadratic equation is correct, you would substitute the solution back into the original equation.

Write down – Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown. For example, if you are given a graph of a function and asked to write down the x-intercept, you simply look at where the function crosses the x-axis and note the value of x at that point. No additional calculations or working are needed.

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