Move the a, b, and c sliders to transform the graph.

- What happens to the graph as the value of a changes?
- What happens to the graph as the value of b changes?
- What happens to the graph as the value of c changes?

The standard form of a quadratic equation, where a,\,b,\, and c are real numbers is:

\displaystyle y=ax^2+bx+c

\bm{a}

scale factor

\bm{b}

linear coefficient

\bm{c}

y-value of the y-intercept

The **standard form of a quadratic equation** allows us to quickly identify the y-intercept and whether the parabola opens up or down.

The coordinates of the **vertex** are: \left(-\dfrac{b}{2a},\,f\left(-\dfrac{b}{2a}\right)\right)

We can substitute the x-coordinate of the vertex into the original equation in order to find the y-coordinate of the vertex.

For example, if we have the function g(x)=3x^2 + 12x - 15 where a=3,\,b=12,\, and c=-15 we can start by finding the x-coordinate:

\displaystyle x | \displaystyle = | \displaystyle -\dfrac{b}{2a} | Equation for the x-coordinate of the vertex |

\displaystyle = | \displaystyle -\dfrac{12}{2(3)} | Substitute a=3 and b=12 | |

\displaystyle = | \displaystyle -\dfrac{12}{6} | Evaluate the multiplication | |

\displaystyle x | \displaystyle = | \displaystyle -2 | Evaluate the division |

We can substitute the x-coordinate of the vertex into the original equation in order to find the y-coordinate of the vertex.

\displaystyle g(x) | \displaystyle = | \displaystyle 3x^2 + 12x - 15 | Original function |

\displaystyle = | \displaystyle 3\left(-2\right)^2 + 12\left(-2\right) - 15 | Substitute x=-2 | |

\displaystyle = | \displaystyle 3\left(4\right) + 12\left(-2\right) - 15 | Evaluate the exponent | |

\displaystyle = | \displaystyle 12 - 24 - 15 | Evaluate the multiplication | |

\displaystyle = | \displaystyle -27 | Evaluate the subtraction |

The coordinates of the vertex of g(x) are \left(-2,\,-27\right). We can confirm this by looking at the graph:

We can also see here that the **axis of symmetry** is the line:

x=-\dfrac{b}{2a}

The axis of symmetry always passes through the vertex.

For the quadratic function y=3x^2-6x+8:

a

Identify the axis of symmetry.

Worked Solution

b

State the coordinates of the vertex.

Worked Solution

c

State the coordinates of the y-intercept.

Worked Solution

d

Draw a graph of the corresponding parabola.

Worked Solution

Naomi is playing a game of Kapucha Toli, where to start a play, a ball is thrown into the air. Naomi throws a ball into the air from a height of 6 feet, and the maximum height the ball reaches is 12.25 feet after 1.25 seconds.

a

Sketch a graph to model the height of the ball over time.

Worked Solution

b

Predict when the ball will be 3 feet above the ground.

Worked Solution

c

Write a quadratic equation in standard form to model the situation.

Worked Solution

Write the standard form equation of the function shown on the graph.

Worked Solution

The whale jumps out the water at 3 seconds and reenters the water after 6.5 seconds. The whale reaches a maximum height of 49 feet after 4.75 seconds. Determine the equation in standard form that models the whale’s jump.

Worked Solution

Idea summary

The **standard form of a quadratic equation** highlights the y-intercept of a quadratic function.

\displaystyle y=ax^2+bx+c

\bm{a}

scale factor

\bm{b}

linear coefficient

\bm{c}

y-value of the y-intercept

The **axis of symmetry** is the line:

x=-\dfrac{b}{2a}

The axis of symmtetry is also the x-coordinate of the vertex. To find the y-coordinate, you substitute the x-coordinate back into the original function. Therefore, the coordinates of the vertex are: \left(-\dfrac{b}{2a},\,f\left(-\dfrac{b}{2a}\right)\right)