For the following examples:
1. Write the equation in standard form and identify a, b, and c.
2. Determine whether the parabola opens up or down.
3. Find the vertex and determine whether it is the maximum or minimum point.
4. Make a rough sketch of the parabola.
• Example 1
• Example 2
y = 2x2 – 4x – 5
1. Standard form and Identify a, b, and c

This equation is already in standard form.

y = 2x2 – 4x – 5

a = 2, b = –4, c = –5

2. Direction of opening

Since a is positive, this parabola opens up.

3. Coordinates of the vertex

x = = = = 1
To find the y-coordinate, substitute the x value of 1 into the quadratic equation and solve
for y.

y = 2x2 – 4x – 5
y = 2(1)2 – 4(1) – 5
y = 2 – 4 – 5
y = –7

The vertex is (1, –7).
Because this parabola opens up, the vertex is the minimum point.

4. Rough sketch of the parabola.

y + 1 = –x2 + 4x
1. Standard form and Identify a, b, and c

y + 1 = –x2 + 4x
–1   –1

y = –x2 + 4x – 1 This is the standard form.

Now for the values of a, b, and c.
y = –1x2 + 4x – 1

a = –1, b = 4, c = –1

2. Direction of opening

Since a is negative, this parabola opens down.

3. Coordinates of the vertex

x = = = = 2
To find the y-coordinate, substitute the x value of 2 into the quadratic equation, and
solve for y.

y = –x2 + 4x – 1
y = –1(2)2 + 4(2) – 1
y = –1(4) + 8 – 1
y = –4 + 8 – 1
y = 3
The vertex is (2, 3).
Because this parabola opens down, the vertex is the maximum point.

4. Rough sketch of the parabola.