![]() That helps you imagine where the graph is going to go. A little counterintuitive but watch out for that. So it's a little counterintuitive here and this is sometimes where students get confused but when you have x minus c, this is a shift to the right and when you have x plus c, this is a shift to the left. I've shifted the entire parabola by three units. ![]() Then if I have the same graph with x plus 3, where does this equal 0? Well, I got to go left three spots to get x equals negative three to output 0. When you graph these parabolas, again, look for where the function outputs zero, where's the minimum of this function going to be? If I have x minus 3, if I plug in 3 for x, then the function outputs 0, you get 0 squared of 2, 0 and so your parabola gets shifted to the right. Take a second if you can, pause the video, and see if you can graph these parabolas and where they live and what they look like. Y equals x minus 3 squared and as a counter, well, let's do x plus 3 squared. Imagine if I saw f of x equals x minus c squared and the other case I want to look at is what if f of x equals x plus c squared? We saw this before as an example. So now let's pick a number and let's let it be positive and I want to consider a parabola of the form, something we just saw. So look out for parabolas in factored form. So x equals 0 in this case, and you get 0 plus 2 quantity squared, which is of course 4. This is when you set the variable x equal to 0, so opposite. As another important piece, you can also find the y-intercept. So looking at where the function outputs zero helps define certain properties of this parabola. So if I plug in negative 2, so let's graph negative 2, and I plug in negative 2 to this function, x plus 2 quantity squared, you get zero on your graph, and then since you're always non-negative, the parabola has a minimum at 0, has a minimum at 0. The smallest you can ever get is zero as your output. Squaring a function leads to a non-negative number. But if you look at it as x plus 2 squared, this is the factored form, there's something about this that helps you see where this parabola lives. If you try to graph x squared plus 4x plus 4, it's a little tricky to see where the min is and where the max is and where this graph lies on the x-y axis. But this also gives us a hint of what's to come. No, the reason for that is if you foil this out, you'll get x squared plus 4x plus 4 and in that case now the degree two is clearly visible, so this is a quadratic function. Some folks look at this and say, x plus 2 is not a line. As another example of a quadratic function, sometimes they come in disguise. So the range can change, but the domain will always be real numbers. It will depend on if the graph goes up or down or where it's located. A quadratic function, as it is a polynomial, will always have a domain of all real numbers and the range will vary. As an example of what you've seen before, you've certainly seen f of x equals x squared, that is just a very specific quadratic function where the leading term a is one and the b and the c are zero, they're both zero. But the general form is ax squared plus bx plus c. They can certainly open upwards, they can open downward and we'll talk about the differences in a second. So we keep hearing quad, it usually means four, but for our cases here, it's going to be the highest exponent that you see is two and this is the shape normally that you know and love, these are your good old parabolas. ![]() This is called a quadratic function, and the reason why it's given a quadratic function, how do you know when something is a quadratic function? The highest term that appears is degree two. As before, f of x equals ax squared plus bx plus c. So first let's define what a quadratic function is. Hello, everyone, and welcome to our lecture on quadratic functions. ![]() Real-life data, exercise sets, and regular assessments help to motivate and reinforce the content in this course, leading to learning and mastery. Students will be presented with not only new ideas, but also new applications of an old subject. Students interested in the natural sciences, computer sciences, psychology, sociology, or similar will genuinely benefit from this introductory course, applying the skills learned to their discipline to analyze and interpret their subject material. This course is designed for all students, not just those interested in further mathematics courses. Concepts of precalculus provide the set of tools for the beginning student to begin their scientific career, preparing them for future science and calculus courses. This is done through studying functions, their properties, and applications to data analysis. This course helps to build the foundational material to use mathematics as a tool to model, understand, and interpret the world around us.
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