2.13.2 Quadratic Graphs
The graph of a quadratic equation is known as a quadratic graph.
An equation of degree \( 2 \)
We can graph a quadratic function by creating a table of \( x \) and \( y \) coordinate values and then plot them on a set of axes.
The shape made by a quadratic graph is known as a parabola.
A quadratic graph can have either a “u” or a “n” shape.
A u-shape will be formed by a quadratic with a positive coefficient of \( x \) squared.
An n-shape quadratic has a negative coefficient of \( x \) squared.
A quadratic function will always intersect the \( y \)-axis.
It may or may not cross the \( x \)-axis.
The point or points where the curve crosses the \( x \)-axis are called roots.
Minimum and maximum points are examples of turning points.
We often only require a sketch to be drawn, showing just the key features.
The most important features of a quadratic are
Its overall shape, which is either a u-shape or an n-shape
Its \( y \)-intercept
The \( y \)-intercept of \( y=ax^2+bx+c \), will be \( 0, c \)
Its \( x \)-intercept(s), also known as its roots
The roots will be the solutions to \( ax^2+bx+c=0 \)
Use factorisation, completing square or quadratic formulas to find roots.
Its turning point
Use differentiation to find turning point
Solve \( \frac{dy}{dx}=0 \) , it gives the \( x \)-coordinate of the turning point
Substitute in the quadratic equation to find \( y \)-coordinate
Worked example:
On the diagram,Sketch the graph of \( y=(x-1)^2 \) .
As it’s a quadratic equation with a positive coefficient of \( x \) squared
So, it will be a “U-shaped” graph
Now put \( x=0 \) to find the point where the graph crosses the y-axis
\( y=(0-1)^2 \)
\( y=1 \)
Put \( y=0 \) to find the point or points where the graph crosses the x-axis
\( 0=(x-1)^2 \)
\( 0=x-1 \)
\( x=1 \)
Because there is only one answer, the graph only touches the \( x \)-axis and does not traverse it.
Draw a symmetrical shape that goes through \( (0, 1) \) and touches the parabola \( (1, 0) \).
Test yourself
Question 1:
Find the turning point of \( y=x^2+4x-3 \) by completing the square.
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Question 2:
Factorize \( 24+5x-x^2 \).
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The diagram shows a sketch of \( y=24+5x-x^2 \).
Work out the values of \( a, b \) and \( c \) .
[3]
Question 3:
Write \( x^2+10x+14 \) in the form \( x+a^2+b \) .
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Sketch the graph of \( y=x^2+10x+14 \), indicating the coordinates of the
turning point.
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Question 4:
Sketch the graph of \( y=x^2-6x+13 \) showing the y intercept and the turning point .
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