The following graph shows bakery profits from bread sales. How do you determine the equation of the quadratic relation from its graph?

5.4 Quadratic Models Using Vertex Form

We know vertex so use “vertex form”y = a(x – h)2 + k - vertex is (2, 9) = (h, k)y = a(x – 2)2 + 9 To find “a” choose any point on the graph (0, 5)5 = a(0 – 2)2 + 9 5 = a(-2)2 + 9 5 = 4a + 95 - 9 = 4a + 9 - 9-4 = 4aaa = -1So y = -(x – 2)2 + 9

Determine Equation from Graph

The amount of gasoline that a car consumes depends on its speed. Below is the relationship between speed and fuel consumption for a particular car.

Example

Speed (km/h) 10 20 30 40 50 60 70 80 90 100 110 120

Gas Consumed (L/100km)

9.2 8.1 7.4 7.2 6.4 6.1 5.9 5.8 6.0 6.3 7.5 8.4

Example

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

7

8

9

10

Gas Consumed vs Speed

Speed (km/h)

Gas (

L/1

00km

)Speed (km/h) 10 20 30 40 50 60 70 80 90 100 110 120

Gas Consumed (L/100km)

9.2 8.1 7.4 7.2 6.4 6.1 5.9 5.8 6.0 6.3 7.5 8.4

We can see the general shape of a parabola here, but it’s not perfect

Why? Because these is modeling a real-life situation with real data, not perfect data

Example

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

7

8

9

10

Gas Consumed vs Speed

Speed (km/h)

Gas (

L/1

00km

)

The curve that models this data is: y = 0.0009(x – 75)2 + 5.8 Not pretty, but it’s accurate and realistic

Example

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

7

8

9

10

Gas Consumed vs Speed

Speed (km/h)

Gas (

L/1

00km

)

If you know the coordinates of the vertex (h, k) and one other point on a parabola, you can determine the equation of the relation using y = a(x – h)2 + k

In Summary…