In an earlier lesson, we saw that an equation such as can model the height of an object thrown upward from a height of 10 feet with a vertical velocity of 78 feet per second.

1. Is the expression written in standard form? Explain how you know.
2. Jada said that the equation also defines the same function, written in factored form. Show that Jada is correct.
3. Here is a graph representing both and .

   

   Graph of non linear function, origin O. Horizontal axis from 0 to 5 by 1’s. Vertical axis from 0 to 140 by 20’s. Line starts at 0 comma 10, increases until about 2 point 5 comma 105 then decreases until 5 comma 0. Passes through 1 comma 72 and 3 comma 100.

   1. Identify or approximate the vertical and horizontal intercepts.
   2. What do each of these points mean in this situation?

Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.

1. Identify the -intercepts and the -intercept of each graph.

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 4 to 4, by 2’s. Line passes through negative 4 comma 3, negative 3 comma 0, negative 2 comma negative 1, negative 1 comma 0, and 0 comma 3.

   -intercepts:

   -intercept:

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 4 to 4, by 2’s. Line passes through 0 comma 4, 1 comma 0, 2 comma negative 2, 3 comma negative 2, 4 comma 0, and 5 comma 4.

   -intercepts:

   -intercept:

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 10 to 2, by 2’s. Line passes through negative 3 comma 0, negative 2 comma negative 5, 0 comma negative 9, 1 comma negative 8, and 3 comma 0.

   -intercepts:

   -intercept:

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 8 to 2, by 2’s. Line passes through 0 comma 0, 1 comma negative 4, 2 comma negative 6, 3 comma negative 6, 4 comma negative 4, and 5 comma 0.

   -intercepts:

   -intercept:

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 2 to 6, by 2’s. Line passes through 0 comma 0, 1 comma 4, 2 comma 6, 3 comma 6, 4 comma 4, and 5 comma 0.

   -intercepts:

   -intercept:

   Function

   

   Graph of non linear function, origin O. Horizontal axis from negative 6 to 4, by 2’s. Vertical axis from negative 2 to 6, by 2’s. Line passes through negative 4 comma 4, negative 2 comma 0, and 0 comma 4

   -intercepts:

   -intercept:

2. What do you notice about the -intercepts, the -intercept, and the numbers in the expressions defining each function? Make a couple of observations.

3. Here is an expression that models function , another quadratic function: . Predict the -intercepts and the -intercept of the graph that represent this function.