1. For example: Does the function
How do you determine whether a quadratic function in standard form has a minimum value or a maximum value?
For example: Does the function
f(x)= -2x2+3x-4 have a minimum or maximum value?
LearnZillion Notes:

2. LearnZillion Notes:

3. x f(x) -4 16 -3 9 -2 4 -1 1 2 3 LearnZillion Notes:2 3
LearnZillion Notes:
You already know how to graph a function by making a table of values.
Jon- should I use f(x) or y in the table of values???

4. f(x) = 2x2+4x-5 f(x) = 5x2-2x LearnZillion Notes:
You already know what the standard form of a quadratic function looks like, and that standard form can have two terms or even just one term if the other terms equal zero.
You also know that the leading coefficient of a quadratic function is the coefficient of the squared term (term with highest degree). For example the leading coefficient is 3 for this function.
And if that the leading coefficient just has a negative sign it means that the leading coefficient is negative 1.

5. LearnZillion Notes:
The shape of graph of the function f(x)=x^2 is a u-shaped curve called a parabola.
The turning point of the curve, the bottom of the “U”, is called the vertex of the graph.
Does this function have a maximum value? No, as we enter larger and larger values for x greater than zero, we see that the graph continues to grow in the positive direction. The same thing happens for values of x less than zero. There is no maximum value the function can have because there will always be a larger value of x that we can use to find a larger value of the function.
Does this function have a minimum value? Yes, we can see that there is no value of x that can possibly make the value of the function less than zero. The min. value of this function is y=0.

6. LearnZillion Notes:
Now lets look at the graph of the function f(x)=-x^2. We see that it is also a parabola and that the vertex is (0,0).
What is different about the graph of this function? Does this function have a minimum value? No, as we enter larger and larger values for x great than zero, we see that the graph continues grow in the negative direction. The same thing happens for smaller and smaller values of x less than zero. There is no minimum value the function can have because there will always be a larger value of x that we can use to find a smaller value of the function.
We see, however, that this function does have a maximum value. There is no value of x that can possibly make the value of the function less than zero. The min. value of this function is y=0.

7. LearnZillion Notes:
Now take a look at the graph of these two functions side by side. What do you see is the main difference between the expressions that define these two functions?
The only difference between the two expressions is that the leading coefficient of the first function is positive 1 and the leading coefficient for the second function is negative one. Notice the first graph opens up and has a minimum value while the second graph opens down and has a maximum value.
Let’s look at a few more examples to see if we might be able to discover a connection between the sign of the leading coefficient and the shape of the graph.

8. LearnZillion Notes:
Here we have a graph with a leading coefficient of positive one. What do you notice about the shape of the graph? This graph opens upwards and thus has a minimum value at the vertex.
What happens to the graph if we have a very similar function, but the coefficient is negative one. What do you notice? The graph opens downwards so it has a maximum value at the vertex.
Let’s take a look at one more set of graphs to see if you can predict whether a function has a maximum or minimum value at the vertex.

9. Notes:
Take a look at these two functions. Which one do you predict will have a maximum value and which will have a minimum value? Let’s look at the graphs to see if your predictions are right.
f(x)= -2x2+3x-4 has a negative leading coefficient so it opens downward and has a maximum value at the vertex.
f(x)= 3x2+2x+1 has a positive leading coefficient so it opens upward and has a minimum value at the vertex.
So in general, If a function’s leading coefficient is positive it will open upward and have a minimum value at its vertex. If its leading coefficient is negative, it will open downward and have a maximum value at its vertex.

10. -2 must mean function has a minimum value
f(x) = -2x2 +x -3
-2 must mean function has a minimum value
LearnZillion Notes:
A common mistake is to think that if a function’s leading coefficient is negative it must have a minimum value.
This might seem to make sense since we tend to associate negative numbers with minimal value in general. However, we need to remember what the graph of the function looks like. A negative leading coefficient will make a graph open downward and the vertex will therefore be the maximum value of the function.
The leading coefficient of -2 tells us this function has a maximum value, not a minimum value.

11. LearnZillion Notes:
--This is the lesson conclusion. On this slide you’ll change your original lesson objective to past tense and explain what the student has just learned. You can retype it here or you can delete the text on this slide and then just copy and paste the text box from the original Lesson Objective slide and then edit it to make it past tense!

12. LearnZillion Notes:
The leading coefficient is positive so the function will have a minimum value.
We can confirm this is true by taking a quick look at the graph of the function.

13. LearnZillion Notes:
The leading coefficient is positive so the function will have a minimum value.
We can confirm this is true by taking a quick look at the graph of the function.

14. The function f(x)= -x2 + 2x +3 can be
rewritten as f(x)= -(x-3)(x+1) in factored form
and as f(x) = -(x+1) in vertex form
Does the form the equation is written in affect whether the function will have a maximum or minimum value? Explain.
LearnZillion Notes:

15. Determine whether the function
f(x)= -5x2 + 2x + 13 has a maximum or minimum value?
Determine whether the function
f(x)= 2x2 + x - 4 has a maximum or minimum value?
LearnZillion Notes:
--”Quick Quiz” is an easy way to check for student understanding at the end of a lesson. On this slide, you’ll simply display 2 problems that are similar to the previous examples. That’s it! You won’t be recording a video of this slide and when teachers download the slides, they’ll direct their students through the example on their own so you don’t need to show an answer to the question.