How can you find the coordinates of the turning point of a quadratic graph?

To find the coordinates of the turning point of a quadratic graph, you can either: complete the square differentiate and set the derivative equal to zero to find the x -coordinate

True or False? You should always use a ruler when plotting the graph of a function.

False. You should only use a ruler if a graph is linear (and for drawing the axes if they are not given). For curves, draw a single smooth freehand curve .

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Graphs of Functions (Edexcel IGCSE Maths A)

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Why does the graph of $y = \frac{1}{x}$ not touch the y-axis?

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Why does the graph of $y = \frac{1}{x}$ not touch the y-axis?
The graph of $y = \frac{1}{x}$ not touch the y-axis because on the y-axis, $x = 0$ . You cannot divide by zero therefore the graph does not have any values on the y-axis.
What is an asymptote?
An asymptote is a line on a graph that a curve gets closer and closer to but never touches.
These may be horizontal or vertical.
E.g. $y = \frac{1}{x}$ has asymptotes at $y = 0$ and $x = 0$ .
How many turning points does the graph of a cubic have?
The graph of a cubic has two turning points; a minimum and a maximum.
However, note that $y = \pm x^{3}$ does not have any turning points.
Will the vertex of the graph $y = - x^{2} + x + 5$ be a maximum or a minimum point?
The vertex of the graph $y = - x^{2} + x + 5$ will be a maximum point.
The vertex of a quadratic graph will be a maximum point if the coefficient of $x^{2}$ is negative. The graph is n-shaped.
Where does the graph of $y = 3 (x - 1) (x + 2)$ cross the x-axis?
The graph of $y = 3 (x - 1) (x + 2)$ crosses the x-axis at $(1, 0)$ and $(- 2, 0)$ .
To find these roots, make each factor equal to zero to find the x-coordinate:
- $x - 1 = 0$ gives $x = 1$
- $x + 2 = 0$ gives $x = - 2$
How can you find the coordinates of the turning point of a quadratic graph?
To find the coordinates of the turning point of a quadratic graph, you can either:
- complete the square
- differentiate and set the derivative equal to zero to find the x-coordinate
True or False?
The turning point of $y = 4 - {(x - 1)}^{2}$ is at the point $(4, 1)$ .
False.
The turning point of $y = 4 - {(x - 1)}^{2}$ is not at the point $(4, 1)$ .
The x-coordinate is the value that makes the squared bracket equal to zero.
The coordinates should be $(1, 4)$ .
True or False?
You should always use a ruler when plotting the graph of a function.
False.
You should only use a ruler if a graph is linear (and for drawing the axes if they are not given).
For curves, draw a single smooth freehand curve.
How would you find the y-intercept of a graph using its equation?
To find the y-intercept of a graph, you would substitute $x = 0$ into the equation.
True or False?
The solutions to $x^{3} - 4 x = 0$ are the value(s) where the graph of $y = x^{3} - 4 x$ crosses the y-axis.
False.
The solutions to $x^{3} - 4 x = 0$ are not the value(s) where the graph of $y = x^{3} - 4 x$ crosses the y-axis.
$x^{3} - 4 x = 0$ when $y = 0$ which is the x-axis. Therefore the solutions are the values where the graph crosses the x-axis.
The solutions of $x^{3} + x^{2} - 3 = x - 2$ are the x values of the intersections between $y = x^{3} + x^{2} - 3$ and which other graph?
The solutions of $x^{3} + x^{2} - 3 = x - 2$ are the x values of the intersections between $y = x^{3} + x^{2} - 3$ and $y = x - 2$ .
True or False?
The x values of the intersections of the two graphs $y = x + 1$ and $y = x^{2} + 5 x + 4$ are the solutions of $x^{2} + 4 x + 3 = 0$ .
True.
The x values of the intersections of the two graphs $y = x + 1$ and $y = x^{2} + 5 x + 4$ are the solutions of $x^{2} + 4 x + 3 = 0$ .
Set the equations equal to each other and rearrange: $x^{2} + 5 x + 4 = x + 1$ .
What is the graph of $y = \sin x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \sin x$ for $- 360 ° \leq x \leq 360 °$ is:
What is the graph of $y = \cos x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \cos x$ for $- 360 ° \leq x \leq 360 °$ is:
What is the graph of $y = \tan x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \tan x$ for $- 360 ° \leq x \leq 360 °$ is:
True or False?
The point $(0, 0)$ lies on the graph $y = \sin x$ .
True.
The point $(0, 0)$ lies on the graph $y = \sin x$ .
What is the y-intercept of the graph $y = \cos x$ ?
The y-intercept of the graph $y = \cos x$ is $(0, 1)$ .
What is the minimum y value of the graph $y = \sin x$ ?
The minimum y value of the graph $y = \sin x$ is -1.
True or False?
The graph $y = \cos x$ repeats itself every 180°.
False.
The graph $y = \cos x$ does not repeat itself every 180°.
It repeats itself every 360°.
True or False?
The graph $y = \tan x$ repeats itself every 180°.
True.
The graph $y = \tan x$ repeats itself every 180°.
The graph $y = \sin x$ repeats itself every how many degrees?
The graph $y = \sin x$ repeats itself every 360°.
True or False?
The maximum y value on the graph $y = \tan x$ is 1.
False.
The maximum y value on the graph $y = \tan x$ is not 1.
The graph $y = \tan x$ does not have a maximum value.
How would you use a graph of $y = \sin x$ to find the solutions of $\sin x = 0.5$ for $0 ° \leq x \leq 360 °$ ?
To find the solutions of $\sin x = 0.5$ for $0 ° \leq x \leq 360 °$ using the graph $y = \sin x$ :
- calculate one solution using inverse trig $x = \sin^{- 1} (0.5)$
- draw the horizontal line $y = 0.5$
- use the symmetry of the graph to find the other solution
True or False?
After finding the first solution for an equation involving the cosine function, you can find another solution by subtracting the first solution from 180º.
False.
After finding the first solution for an equation involving the cosine function, you can find another solution by subtracting the first solution from 360º.
What angle should you add to or subtract from a first solution to find another solution for an equation involving the tangent function?
If you know a first solution for an equation involving the tangent function, you can add to or subtract 180º from it to find another solution .

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