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 .

What is meant by a tangent to a graph?

A tangent is a line that touches a curve at a point (and does not cross it).

What is meant by the gradient of a curve at a point ?

The gradient of a curve at a point is defined to be the same as the gradient of the tangent to the curve at that point.

True or False? Drawing a tangent line to a curve at a point will always give the exact gradient of the curve at that point.

False. Drawing a tangent line to a curve at a point will not always give the exact gradient of the curve at that point. A tangent line is drawn by eye so it will give an estimate of the gradient. Differentiation can be used to find the exact equation of the tangent.

How do you estimate the gradient of a curve at a point ?

To find an estimate for the gradient of a curve at a point : Draw a tangent to the curve at the point. Find the gradient of the tangent using Gradient = rise ÷ run or difference in y ÷ difference in x . The gradient of the tangent will give the estimate for the gradient of the curve at the point.

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Further Graphs & Tangents (Cambridge (CIE) IGCSE Maths)

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FrontTypes of Graphs
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.
True or False?
$y = a k^{x}$ is an example of an exponential graph.
True.
$y = a k^{x}$ is an example of an exponential graph; it has a variable power.
E.g. $y = 2 \times 3^{x}$
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 $x = 0$ .
True or False?
The graph of $y = x^{\frac{1}{2}}$ is never below the $x$ -axis.
True.
The graph of $y = x^{\frac{1}{2}}$ is never below the $x$ -axis.
It is equivalent to $y = \sqrt{x}$ and this means the positive root of $x$ .
So it is on the $x$ -axis when $x = 0$ , and above the $x$ -axis for all other (positive) values of $x$ .
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 meant by a tangent to a graph?
A tangent is a line that touches a curve at a point (and does not cross it).
What is meant by the gradient of a curve at a point?
The gradient of a curve at a point is defined to be the same as the gradient of the tangent to the curve at that point.
True or False?
Drawing a tangent line to a curve at a point will always give the exact gradient of the curve at that point.
False.
Drawing a tangent line to a curve at a point will not always give the exact gradient of the curve at that point.
A tangent line is drawn by eye so it will give an estimate of the gradient.
Differentiation can be used to find the exact equation of the tangent.
How do you estimate the gradient of a curve at a point?
To find an estimate for the gradient of a curve at a point:
- Draw a tangent to the curve at the point.
- Find the gradient of the tangent using
  Gradient = rise ÷ run
  or difference in y ÷ difference in x.
- The gradient of the tangent will give the estimate for the gradient of the curve at the point.

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