Quadratic Graphs

Quadratic Graphs

Quadratic graphs

What is a quadratic graph?

• A quadratic graph has the form $y=a{x}^2+bx+c$ 

  • where $a$ is not zero

What does a quadratic graph look like?

• A quadratic graph is a smooth curve with a vertical line of symmetry

  • A positive number in front of $x^2$ gives a u-shaped curve

  • A negative number in front of $x^2$ gives an n-shaped curve

• The shape made by a quadratic graph is known as a parabola

• A quadratic graph will always cross the $y$-axis

• A quadratic graph intersects the $x$-axis twice, once, or not at all

  • The x values where the graph intersects the $x$-axis are called the roots

• If the graph is a u-shape, it has a minimum point

• If the graph is an n-shape, it has a maximum point

• Minimum and maximum points are both examples of turning points

  • A turning point can also be called a vertex

How do I sketch a quadratic graph?

• It is important to know how to sketch a quadratic curve

  • A simple drawing showing the key features is often sufficient

  • (For a more accurate graph, create a table of values and plot the points)

• To sketch a quadratic graph:

  • First sketch the $x$ and $y$-axes

  • Identify the $y$-intercept and mark it on the $y$-axis

    • The $y$-intercept of $y=a{x}^2+bx+c$ will be $\left(0, c\right)$

    • It can also be found by substituting in $x=0$

  • Find all root(s) (0, 1 or 2) of the equation and mark them on the $x$-axis

    • The roots will be the solutions to $y=0$; $a{x}^2+bx+c=0$

    • You can find the solutions by factorising, completing the square or using the quadratic formula

  • Identify if the number $a$ in $a{x}^2+bx+c$ is positive or negative

    • A positive value will result in a u-shape

    • A negative value will result in an n-shape

  • Sketch a smooth curve through the $x$ and $y$-intercepts

    • Mark on any axes intercepts

    • Mark on the coordinates of the maximum/minimum point if you know it

How do I find the coordinates of the turning point by completing the square?

• The coordinates of the turning point (vertex) of a quadratic graph can be found by completing the square

• For a quadratic graph written in the form $y=a{\left(x-p\right)}^2+q$

  • the minimum or maximum point has coordinates $\left(p, q\right)$

• Beware: there is a sign change for the $x$-coordinate

  • A curve with equation $y={\left(x-3\right)}^2+2$, has a minimum point at $\left(3, 2\right)$

  • A curve with equation $y={\left(x+3\right)}^2+2$, has a minimum point at $\left(-3, 2\right)$

• The value of $a$ does not affect the coordinates of the turning point but it will change the shape of the graph

  • If it is positive, the graph will be a u-shape

    • The curve $y=5{\left(x-3\right)}^2+2$ has a minimum point at $\left(3, 2\right)$

  • If it is negative, the graph will be an n-shape

    • The curve $y=-8{\left(x-3\right)}^2+2$ has a maximum point at $\left(3, 2\right)$

How do I find the coordinates of the turning point using differentiation?

• The coordinates of the turning point (maximum/minimum) of a quadratic can be found through differentiation

• To find the coordinates of the turning point

  • Differentiate the quadratic equation $y=a{x}^2+bx+c$

    • This will give you $\frac{\mathrm{d}y}{\mathrm{d}x}$

  • Set $\frac{dy}{dx}=0$  and solve for $x$

    • The solution will be the $x$-coordinate of the turning point

  • Substitute the $x$ value into $y=a{x}^2+bx+c$

    • This will give you the $y$-coordinate of the turning point

How do I find the equation of a quadratic from its graph?

• If the vertex and one other point are known

  • Use the form $y=a{\left(x-p\right)}^2+q$ to fill in $p$ and $q$

    • The vertex is at $\left(p, q\right)$

  • Then substitute in the other known point $\left(x, y\right)$ to find $a$

• If the roots ($x$-intercepts) and one other point are known

  • Use the form $y=a\left(x-x_1\right)\left(x-x_2\right)$ to fill in $x_1$ and $x_2$

    • The roots are at $\left(x_1 , 0\right)$ and $\left(x_2 , 0\right)$

  • Then substitute in the other known point $\left(x, y\right)$ to find $a$

• If $a=1$ then you only need either the vertex or the roots