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[h] IB Mathematics AA HL Flashcards- Key features of graphs

[q] Key features of graphs

You may well be asked to draw or sketch the graphs of functions. Drawing should be as accurate as possible, including all relevant graph features. Sketching should just give a general idea of the shape, with some features, depending on how the question is phrased, and what type of function is asked for.

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FEATURES

Here we can define the main types of graph features:

– Maximum/Minimum values: ‘Turning points’ of the graph: max or min.

– Intercepts: Where the graph crosses the x-axis or the y-axis.

– Line of symmetry: Mainly regarding quadratics, self-explanatory.

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– Vertex: Max. or min. points for quadratic function.

– Zeros, Roots: Where \(y = 0\), i.e.: the x-intercepts.

– Asymptotes: A line that a graph will get infinitely close to but will never touch, as the x or y tends to infinity.

– Intersection points: The point where two lines cross (intersect).

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FEATURES ON A GRAPH

To visualize the above features:

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GDC

To graph almost any function, we press:

– TI-nspire →

\( \boxed{B: \text{GRAPH}} \) → \( \boxed{\text{TAB}} \) → type function → \( \boxed{\text{ENTER}} \)

– TI-84 →

\( \boxed{Y=} \) → type function → \( \boxed{\text{GRAPH}} \)

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FINDING FEATURES w/GDC ➔ You can find the features previously defined by using your GDC’s ‘analyse graph’ capabilities:

TI-nspire ➔ MENU ➔ 6: ANALYSE GRAPH… // TI-84 ➔ 2ND ➔ CALC

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FINDING FEATURES ➔ We can find many of these manually:

– Turning Points ➔ You can use calculus (see topic 5), or for quadratics, you can use \(x = \frac{-b}{2a}\) (see 2.6 for A&A, 2.5 for A&I), otherwise use a GDC.

– Intercepts ➔ To be on the y-axis, you must have \(x = 0\), so if you plug in \(x = 0\) and solve for \(y\), you have the y-intercept. Reverse this for x-intercept(s).

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– Zeros & Roots ➔ Again, set \(y = 0\) and solve.

– Vertical Asymptotes ➔ At an x value of restricted domain, such as the x value that means you would be dividing by 0.

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– Horizontal Asymptotes ➔ To find what value the graph gets closer and closer to, we can plug in very large values for x. Or use advanced calculus (limits).

– Intersection Points ➔ See 2.7 (set functions equal to each other, and solve)

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E.G. 1 ➔ Draw the graph of \(f(x) = \frac{2}{x – 1} + 1\):

– Divide by zero when \(x = 1\), so V.A. at \(x = 1\)

– As \(x \rightarrow \infty\), \(f(x) = 0 + 1 = 1\), so H.A. at \(x = 1\)

– Plug in \(x = 0\): \(f(0) = \frac{2}{0 – 1} + 1 = -1\), y-int. at \((0, -1)\)

– Set \(f(x) = 0\): \(0 = \frac{2}{x – 1} + 1 \Rightarrow x = -1\), x-int. at \((-1, 0)\)

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