Where does a positive cubic graph start and end?

A positive cubic comes up from the bottom left and leaves at the top right . A negative cubic does the opposite, starting at the top left and ending at the bottom right.

True or False? Solutions read from a graph are exact.

False. You are reading a position off a scale by eye, so the accuracy is limited by the drawing. Such answers are given to a stated accuracy, like 1 decimal place, rather than exactly.

GCSEFurther MathsAQAFlashcards2. Algebra & FunctionsShapes of Graphs

Shapes of Graphs (AQA GCSE Further Maths): Flashcards

Exam code: 8365

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FrontQuadratic & Polynomial Graphs
What decides whether a quadratic graph is a u-shape or an n-shape?

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Cards in this collection (22)

What decides whether a quadratic graph is a u-shape or an n-shape?
The sign of the coefficient of $x^{2}$ : positive gives a u-shape, negative gives an n-shape.
A u-shape has a minimum point and an n-shape has a maximum point.
The curve itself is called a parabola.
Complete the $y$ -intercept of $y = a x^{2} + b x + c$ :
$(0,______)$
The $y$ -intercept is at:
$(0, c)$
Substituting $x = 0$ kills the $a x^{2}$ and $b x$ terms, leaving just the constant.
How do you find the roots of a quadratic graph?
Set $y = 0$ and solve $a x^{2} + b x + c = 0$ , since the roots are where the curve crosses the $x$ -axis.
There may be 2, 1 or 0 roots, so a quadratic graph need not cross the $x$ -axis at all.
The graph of $y = x^{2} - 6 x + 13$ has a minimum point at $(3, 4)$ . How many roots does it have?
The graph has no roots, because the lowest point of the curve is 4 units above the $x$ -axis, so the curve never reaches it.
For a u-shaped graph, where the minimum sits relative to the $x$ -axis tells you at once whether there are 2, 1 or 0 roots.
True or False?
A quadratic graph whose maximum point lies on the $x$ -axis has exactly one root.
True.
The curve touches the axis at that single point and stays below it everywhere else.
$y = - {(x + 2)}^{2}$ is an example, with its maximum at $(- 2, 0)$ .
What four things should you find before sketching a polynomial graph?
The $y$ -axis intercept, the $x$ -axis intercepts (the roots), the turning points, and the overall shape.
Then join them with a smooth curve rather than straight segments.
Where does a positive cubic graph start and end?
A positive cubic comes up from the bottom left and leaves at the top right.
A negative cubic does the opposite, starting at the top left and ending at the bottom right.
Complete the number of turning points a cubic graph can have:
A cubic has $______$ or $______$ turning points.
The completed statement is:
A cubic has 2 or 0 turning points.
A quartic has 3 or 1, and a quadratic always has exactly 1.
Sketching $y = 3 x^{4} + 9 x^{3} + 6 x^{2}$ , why does the curve touch the $x$ -axis at the origin?
Because it factorises to $3 x^{2} (x + 1) (x + 2)$ , so $x = 0$ is a repeated root.
A repeated root makes the curve touch the axis and turn back, while a single root makes it cross.
A cubic has a maximum at $(- 1, 26)$ , a minimum at $(2, - 28)$ and crosses the $y$ -axis at 12. How many times does it cross the $x$ -axis?
The curve crosses three times, because the maximum is above the axis and the minimum is below it.
It must come up to the maximum, down through the axis to the minimum, then up through the axis again.
Define an exponential function.
An exponential is a function where the power is the variable, such as $y = 3^{x}$ .
On this course they take the form $y = a b^{x}$ or $y = a b^{- x}$ , with $b > 0$ .
When is $y = b^{x}$ increasing, and when is it decreasing?
$y = b^{x}$ increases when $b > 1$ and decreases when $0 < b < 1$ .
Changing the power to $- x$ swaps the two, so $y = b^{- x}$ decreases when $b > 1$ .
Why is the $y$ -intercept of $y = a b^{x}$ always $(0, a)$ ?
Because substituting $x = 0$ gives $a \times b^{0}$ , and any positive number to the power 0 is 1.
So for $y = b^{x}$ , where $a$ is 1, the intercept is $(0, 1)$ .
True or False?
An exponential graph eventually crosses the $x$ -axis.
False.
The curve gets closer and closer to the $x$ -axis but never reaches it, because $b^{x}$ is never zero.
That is also why an exponential graph has no maximum or minimum point.
What does the $a$ do in $y = a b^{x}$ ?
The $a$ stretches $y = b^{x}$ vertically by scale factor $a$ .
If $a$ is negative it also reflects the graph in the $x$ -axis, putting the whole curve below it.
The curve $y = a b^{- x}$ passes through $(0, 6)$ and $(2, 0.375)$ . Find $a$ and $b$ .
The $y$ -intercept gives $a = 6$ straight away.
Substituting the second point gives $0.375 = 6 b^{- 2}$ , so $b^{2} = 16$ and $b = 4$ , taking the positive root.
Where are the solutions of $f (x) = 0$ read from a graph?
From the $x$ -axis: they are the $x$ -coordinates where $y = f (x)$ crosses it.
Solutions read off a graph are also called roots.
How do you use the graph of $y = f (x)$ to solve a different equation?
Rearrange the equation you want into the form $f (x) = m x + c$ , where $f (x)$ is exactly the plotted curve.
Then draw the line $y = m x + c$ and read off the $x$ -coordinates where it crosses the curve.
True or False?
Solutions read from a graph are exact.
False.
You are reading a position off a scale by eye, so the accuracy is limited by the drawing.
Such answers are given to a stated accuracy, like 1 decimal place, rather than exactly.
When do you not need to draw any extra line?
When the equation to solve is exactly $f (x) = 0$ , the plotted curve set equal to zero.
The $x$ -axis is already the line $y = 0$ , so its crossings are the solutions.
Complete the link between a graph and the solutions:
The number of times the line crosses the curve is the number of $______$ .
The completed statement is:
The number of times the line crosses the curve is the number of solutions.
Each intersection gives one value of $x$ , so counting the crossings counts the solutions.
The graph of $y = x^{3} + x^{2} - 3 x - 1$ is drawn. What line solves $x^{3} + x^{2} - 4 x = 0$ ?
Adding $x - 1$ to both sides gives $x^{3} + x^{2} - 3 x - 1 = x - 1$ , so draw the line $y = x - 1$ .
Its intersections with the curve give $x = - 2.6$ , $x = 0$ and $x = 1.6$ to 1 decimal place.

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