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Here is a graph of the equation .
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.
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.
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:
What do you notice about the -intercepts, the -intercept, and the numbers in the expressions defining each function? Make a couple of observations.
Here is an expression that models function , another quadratic function: . Predict the -intercepts and the -intercept of the graph that represent this function.
Different forms of quadratic functions can tell us interesting information about the function’s graph. When a quadratic function is expressed in standard form, it can tell us the -intercept of the graph that represents the function.
For example, the graph representing has its -intercept at . This makes sense because the -coordinate is the -value when is 0. Evaluating the expression at gives , which equals 7.
When a function is expressed in factored form, it can help us see the -intercepts of its graph. Let’s look at function , given by and function , given by .
If we graph , we see that the -intercepts of the graph are and . Notice that 1 and 4 also appear in , and they are subtracted from .
Graph of non linear function, origin O. Horizontal axis from negative 2 to 5, by 1’s. Vertical axis from negative 4 to 6, by 2’s. Points plotted at 1 comma 0 and 4 comma 0. Line decreases until about 2 point 5 comma negative 21, then increases.
If we graph , we see that the -intercepts are at and . Notice that 2 and 6 are also in the equation , but they are added to .
Graph of non linear function, origin O. Horizontal axis from negative 8 to 1, by 1’s. Vertical axis from negative 6 to 4, by 2’s. Points plotted at negative 6 comma 0 and negative 2 comma 0. Line decreases until about negative 4 comma negative 4, then increases.
The connection between the factored form and the -intercepts of the graph tells us about the zeros of the function (the input values that produce an output of 0).