As the vertex lies on the axis of symmetry, we know the x-coordinate of the vertex. We can substitute the x-coordinate of the vertex into the original equation to get the y-coordinate of the vertex.

Worked examples

Example 1

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

Identify the axis of symmetry.

Approach

We will use the formula x=-\dfrac{b}{2a}, so we need to identify the values of a and b from the equation.

a=3,b=-6

Solution

\displaystyle x	\displaystyle =	\displaystyle -\dfrac{b}{2a}	Equation for axis of symmetry
\displaystyle x	\displaystyle =	\displaystyle -\dfrac{-6}{2\left(3\right)}	Substitute in the values
\displaystyle x	\displaystyle =	\displaystyle -\dfrac{-6}{6}	Simplify the denominator
\displaystyle x	\displaystyle =	\displaystyle 1	Simplify fully

The axis of symmetry is x=1.

State the coordinates of the vertex.

Approach

Once we have the x-coordinate of the vertex from the axis of symmetry, we can substitute it into y=3x^2-6x+8 and evaluate to get y.

From part (a), we know that the axis of symmetry is x=1 so the x-coordinate of the vertex is x=1.

Solution

\displaystyle y	\displaystyle =	\displaystyle 3x^2-6x+8	Stating the given equation
\displaystyle y	\displaystyle =	\displaystyle 3\left(1\right)^2-6\left(1\right)+8	Substituting x=1
\displaystyle y	\displaystyle =	\displaystyle 3-6+8	Simplify
\displaystyle y	\displaystyle =	\displaystyle 5	Simplify fully

The vertex is \left(1,5\right).

State the coordinates of the y-intercept.

Approach

Since the y-intercept occurs when x=0, we can substitute x=0 into the equation and evaluate y.

When we are given an equation in standard form, the y-value of the y-intercept will be y=c.

Solution

In this case, the value of c in the equation is 8.

So we have that the coordinates of the y-intercept are \left(0, 8\right).

Draw a graph of the corresponding parabola.

Approach

We have all the key features we need to create a graph. For more accuracy, we can use the axis of symmetry and y-intercept to find another point. This point will be a reflection of the y-intercept across the axis of symmetry.

We know that the parabola will open upwards because a>0.

Solution

-1

Axis of symmetry: x=1

Vertex: \left(1,5\right)

y-intercept: \left(0,8\right)

Another point: \left(2,8\right)

Reflection

From the graph we can identify that the vertex form of the equation would be:

y=3\left(x-1\right)^2+5

Example 2

For the quadratic function y = -x^2 + 7x - 10:

State the coordinates of the y-intercept.

Approach

Since the y-intercept occurs when x=0, we can substitute x=0 into the equation and determine the value of y.

When we are given an equation in standard form, the y-value of the y-intercept will be y=c.

Solution

In this case, the value of c in the equation is -10.

So we have that the coordinates of the y-intercept are \left(0, -10\right).

State the coordinates of the x-intercept(s).

Approach

Since the x-intercept(s) occur when y = 0, we can substitute y = 0 into the equation and solve for the values of x.

Since the equation is given in standard form, this will involve factoring the quadratic.

Solution

Setting y = 0, we have the equation -x^2 + 7x - 10 = 0. We can now solve this for x as follows:

\displaystyle -x^2 + 7x - 10	\displaystyle =	\displaystyle 0	Substituting y = 0
\displaystyle x^2 - 7x + 10	\displaystyle =	\displaystyle 0	Multiply both sides by -1
\displaystyle \left(x - 2\right)\left(x - 5\right)	\displaystyle =	\displaystyle 0	Factoring the quadratic

The two values of x which satisfy this equation are x = 2 and x = 5. This tells us that the coordinates of the two x-intercepts are \left(2, 0\right) and \left(5, 0\right).

Draw a graph of the corresponding parabola.

Approach

We now know the coordinates of the x- and y-intercepts of the function. We can use these to plot the parabola. For more accuracy, we can also note that the axis of symmetry will occur exactly halfway between the x-intercepts - so in this case, the axis of symmetry will be the line x = \frac{2 + 5}{2} = 3.5.

We also know that the parabola will open downwards because a<0.

Solution

-4

-2

-10

-8

-6

-4

-2

y-intercept: \left(0, -10\right)

x-intercepts: \left(2,0\right) and \left(5, 0\right)

Example 3

To start a play in a game of Kapucha Toli, a ball is thrown in the air. The given equation models the height of the ball above the ground, where y represents the height in feet and t represents time in seconds.

y=-5t^2+10t+4

Determine the height that the ball was thrown from.

Approach

When the ball is thrown, t=0. This statement is asking for the value of the y-intercept.

Solution

The y-intercept is y=4.

This means that the ball was thrown from a height of 4 feet above the ground.

Calculate when the ball reached its maximum height.

Approach

The maximum occurs at the vertex. We are asked "when", so we are looking for the time (t-value of the vertex).

Remember that the equation to find the axis of symmetry is x=-\dfrac{b}{2a}.

In this case, x=t, a=-5, and b=10

Solution

Finding the t-coordinate of the vertex:

\displaystyle t	\displaystyle =	\displaystyle -\dfrac{b}{2a}	Equation for axis of symmetry
\displaystyle t	\displaystyle =	\displaystyle -\dfrac{10}{2\left(-5\right)}	Substitution
\displaystyle t	\displaystyle =	\displaystyle -\dfrac{10}{-10}	Simplify
\displaystyle t	\displaystyle =	\displaystyle 1	Simplify fully

The axis of symmetry is t=1.

The ball reached the maximum height after 1 second.

Calculate the maximum height reached by the ball.

Approach

This is asking for the y-coordinate of the vertex. From part (b), we know that the axis of symmetry is t=1, so we want to use this to find the y-coordinate of the vertex.

Solution

\displaystyle y	\displaystyle =	\displaystyle -5t^2+10t+4	Stating the given equation
\displaystyle y	\displaystyle =	\displaystyle -5\left(1\right)^2+10\left(1\right)+4	Substitute t=1
\displaystyle y	\displaystyle =	\displaystyle -5+10+4	Simplify
\displaystyle y	\displaystyle =	\displaystyle 9	Simplify fully

The maximum height reached by the ball is 9 feet.

Outcomes

MA.912.AR.1.3

Add, subtract and multiply polynomial expressions with rational number coefficients.

MA.912.AR.1.7

Rewrite a polynomial expression as a product of polynomials over the real number system.

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

7.05 Quadratic functions in standard form

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Approach

Solution

Approach

Solution

Reflection

Example 2

Approach

Solution

Approach

Solution

Approach

Solution

Example 3

Approach

Solution

Approach

Solution

Approach

Solution

Outcomes

MA.912.AR.1.3

MA.912.AR.1.7

MA.912.AR.3.7

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