Define a local maximum of a graph.

A local maximum is a turning point where the curve stops rising and begins to fall, so it is the highest point in its own neighbourhood. It need not be the highest point on the whole graph, which is exactly what the word local is there to signal.

What key features can the graph analysis menu on a calculator find for you?

It finds the zeros (also called roots), the local maximum and minimum points , and the intersection of two graphs. On some models this menu is called analyze graph , and on others it is reached with the G-Solv button.

Why must you click either side of a point before the calculator will find it?

The calculator searches only inside the interval you have marked, so the two clicks tell it where to look . A graph can have several zeros or several turning points, and this is how you say which one you want.

True or False? To sketch a graph from your calculator you should first make a table of values and plot the points.

False. A sketch needs no plotted points and no table of values at all. You copy the shape as a single smooth freehand curve and mark the key features on it, and you do not use a ruler for a curve.

How can you be sure you have found all the solutions of an equation from a graph?

Count every place the two graphs cross, checking that the viewing window is wide enough to show them all. A curve and a line can meet more than once: a cubic and a straight line may cross at three points, giving three solutions.

Why can the answer to a graphical inequality need two separate ranges?

The two graphs can swap over more than once, so the curve you want on top can be on top in more than one place. A cubic and a straight line can cross three times, which leaves the cubic below the line in two separate stretches.

IGCSEMathsCambridge (CIE)International MathsExtendedFlashcardsCoordinate Geometry & GraphsUsing Calculators for Graphs, Equations & Inequalities

Using Calculators for Graphs, Equations & Inequalities (Cambridge (CIE) IGCSE International Maths: Extended): Flashcards

Exam code: 0607

1/22

0Still learning

Know0

FrontUsing Calculators to Sketch Graphs
Define a local maximum of a graph.

Was this flashcard helpful?

Cards in this collection (22)

Define a local maximum of a graph.
A local maximum is a turning point where the curve stops rising and begins to fall, so it is the highest point in its own neighbourhood.
It need not be the highest point on the whole graph, which is exactly what the word local is there to signal.
What key features can the graph analysis menu on a calculator find for you?
It finds the zeros (also called roots), the local maximum and minimum points, and the intersection of two graphs.
On some models this menu is called analyze graph, and on others it is reached with the G-Solv button.
Why must you click either side of a point before the calculator will find it?
The calculator searches only inside the interval you have marked, so the two clicks tell it where to look.
A graph can have several zeros or several turning points, and this is how you say which one you want.
True or False?
To sketch a graph from your calculator you should first make a table of values and plot the points.
False.
A sketch needs no plotted points and no table of values at all.
You copy the shape as a single smooth freehand curve and mark the key features on it, and you do not use a ruler for a curve.
A calculator can build a table of values once the function has been entered. Complete the three settings it asks you for:
The $______$ value is the first $x$ in the table, the $______$ value is the last one, and the $______$ size is the gap from one $x$ to the next.
The completed settings are:
The start value is the first $x$ in the table, the end value is the last one, and the step size is the gap from one $x$ to the next.
A smaller step gives more rows, and so a finer picture of how the function behaves.
Which axis intercept can you find without using the calculator at all?
The $y$ -intercept, because substituting $x = 0$ into the function gives it straight away.
Some models do have a Y-ICEPT button, but no calculator is needed for this one.
The graph comes out looking almost blank on the calculator screen. What should you do?
Change the viewing window: zoom out, or use the zoom-fit option, which scales the axes to the key features for you.
Some models instead have a V-Window menu where you type in the exact range of $x$ and $y$ values to show.
Why is a calculator especially useful for a function you have never seen before?
You do not need to know the shape in advance, because the calculator draws it from the equation alone.
That is how you would handle something like $y = x^{\frac{1}{2}}$ without ever having met that curve.
How do you solve $f_{1} (x) = f_{2} (x)$ using a graphical method?
Draw $y = f_{1} (x)$ and $y = f_{2} (x)$ on the same axes, then find their points of intersection.
The solutions are the $x$ -coordinates of those points, and not the full coordinate pairs.
How do you put a second graph onto the same axes on a calculator?
Enter it on the next function line down, $f_{2} (x)$ , underneath the first one.
The tab key is often a shortcut for opening that second line, and both curves are then drawn together.
True or False?
Solving $f (x) = 0$ on a graph means finding where the curve crosses the $x$ -axis.
True.
The right-hand side is $y = 0$ , which is the equation of the $x$ -axis itself.
The intersections you are looking for are therefore the curve's $x$ -intercepts, which the calculator calls its zeros.
To solve $x^{2} + 3 x + 1 = 1$ on a calculator, what second graph do you draw?
You draw $y = 1$ , which is a horizontal line cutting the $y$ -axis at 1.
A constant on one side of an equation is always a horizontal line, however complicated the other side happens to be.
A calculator offers two different routes to the solutions of an equation. Complete the sentence naming them:
Finding where two graphs cross is a $______$ method, while bringing every term to one side and using the equation solver is an $______$ method instead.
The completed sentence is:
Finding where two graphs cross is a graphical method, while bringing every term to one side and using the equation solver is an algebraic method instead.
Both routes give the same answers, because the points where the two graphs cross are exactly the roots of the single rearranged equation.
You already have $y = x^{2} + 3 x + 1$ drawn. How can you use it to solve $x^{2} + 3 x - 4 = 0$ ?
Rearrange the new equation so that the graph you already have appears on one side, by adding 5 to both sides.
That turns it into $x^{2} + 3 x + 1 = 5$ , so you need only add the line $y = 5$ rather than redraw anything.
How can you be sure you have found all the solutions of an equation from a graph?
Count every place the two graphs cross, checking that the viewing window is wide enough to show them all.
A curve and a line can meet more than once: a cubic and a straight line may cross at three points, giving three solutions.
What is the first thing to do when solving an inequality with a calculator?
Replace the inequality sign with an equals sign and solve $f_{1} (x) = f_{2} (x)$ first.
Those solutions are the boundaries of the answer, being the values of $x$ at which the two graphs swap over.
How do you decide which parts of the graph satisfy $f_{1} (x) > f_{2} (x)$ ?
Look for the stretches of $x$ where $y = f_{1} (x)$ is the top curve and $y = f_{2} (x)$ is the bottom curve.
Being higher up the page is what being greater means, so those stretches are exactly where the inequality holds.
True or False?
Solving an inequality graphically gives you a list of $x$ -values, just as solving an equation does.
False.
An inequality is satisfied over whole ranges of $x$ , rather than at isolated points.
An answer therefore looks like $x < 2$ or $- 1 \leq x \leq 3$ , describing every value in a stretch and not just a few of them.
The graphs of $y = x^{2} + 3 x + 1$ and $y = 2 x + 1$ cross at $x = - 1$ and $x = 0$ . Solve $x^{2} + 3 x + 1 > 2 x + 1$ .
The quadratic is the top curve outside the two crossing points, because a u-shaped curve dips below a straight line only between them.
The solution is therefore $x < - 1$ or $x > 0$ .
The inequality sign in your answer has to match the one in the question. Complete the rule:
A question written with $<$ or $>$ has an answer written with the same $______$ signs, so the boundary values are $______$ from the set of solutions.
The completed rule is:
A question written with $<$ or $>$ has an answer written with the same strict signs, so the boundary values are excluded from the set of solutions.
With $\leq$ or $\geq$ the boundary values are included instead, and the answer is written with those signs.
Why can the answer to a graphical inequality need two separate ranges?
The two graphs can swap over more than once, so the curve you want on top can be on top in more than one place.
A cubic and a straight line can cross three times, which leaves the cubic below the line in two separate stretches.
Two graphs cross at $x = 1$ and $x = 4$ , and $f_{1} (x) > f_{2} (x)$ has solution $1 < x < 4$ . Solve $f_{1} (x) < f_{2} (x)$ .
The solution is everything that is left over, $x < 1$ or $x > 4$ .
The graphs change places only where they cross, so wherever one of them is not on top the other one must be.

Sign up to unlock flashcards

By signing up you agree to our Terms and Privacy Policy

Previous:Types of GraphsNext:Sketching 2D Inequalities