And so how do I know what the y value is? Well I can figure out, I can substitute back I could draw an axis of symmetry for my parabola. You say okay, if I want to go right in between the two, I want to be two away from each of them. Well let's see, that's four over two, that's just going to be two, so two comma zero. Of negative two and six or the average negative Is just going to be zero, it's going to sit on the x-axis. Well the midpoint, those are just the average of the coordinates. So I'm trying to find the midpoint between the point negative two comma zero and six comma zero. Midpoint between the point, let's use a new color. Six plus negative two isįour divided by two is two. Of six and negative two? Well, you could do that in your head. So what is the midpoint between, or what is the average Symmetry for your parabola is going to sit right between So given this, how do weįigure out the vertex? Well the key idea here is to recognize that your axis of Negative two right over there, and x is equal to six. Is going to intersect the x-axis at x equals Going to be equal to zero, and y would be equal to zero. If x is equal to negative two, then this right over here is Going to be equal to zero, and then y is going to be equal to zero. If x is equal to six, then this right over here is These are the two x values where y will be equal to zero. Or you subtract two from both sides here and you get x is equal to, these cancel out, you get x is equal to negative two. X minus six equal zero? Well you could add six to both sides, you're probably able toĭo that in your head, and you get x is equal to six. Satisfy either of these would make y equal zeroĪnd those would be places where our curve is Or if x plus two is equal to zero, that would also make this equation true. So if x minus six is equal to zero, then that would make this equation true. Well one half is one-half, it's not going to be equal to zero. If I have the product of multiple things and it needs to be equal to zero, the only way that's going to happen is if one or more of these things are going to be equal to zero. One half times x minus six, times x plus two is equal to zero. Have to figure out when, if we want to know when y equals zero, then we have to solve for when does this expression equal zero? So let's just solve the equation. When does y equal zero? Well to solve that we just To get the general shape of this curve which is And then from that, we'llĪctually be able to find the coordinates of the vertex and we're going to be able Times that we're intersecting the x-axis. When does y equal zero? Which are going to be the And the form that it's in, it's in factored form already, it makes it pretty straightforward for us to recognize So let's see if we can find out where this intersects the x-axis. Now a parabola you might remember can intersect the x-axis multiple times. I'm going to get x-squared, plus something, plus something else. Six times x plus two, I'm going to get a quadratic. The key realization here withoutĮven having to do the math is if I multiply this out, if I multiplied x minus We can get the essence of this graph withoutĭoing that much work. Try to connect the curve that connects all of those dots. Try out a bunch of x values and a bunch of y values and There's many different ways that you could attempt to graph it. So like always, pause this video and take out some graph paper or even try to do it onĪ regular piece of paper and see if you can graph this equation. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.To graph the equation y is equal to one-half times x minus six times x plus two. Previous Lesson Table of Contents Next Lesson This will produce the general form which isĪnd the vertex is found by substituting the x-value of the axis of symmetry into the function to get the y-value. General Form of a Quadratic Functionīoth the standard form and vertex form of a quadratic function can be simplified by multiplying out the expression. The vertex is on the axis of symmetry, so it can be found by substituting the x-coordinate of the axis of symmetry into the original function to find the y-value. Intercept Form of a Quadratic Functionīecause of symmetry, the axis of symmetry is halfway between the x-intercepts. This lesson introduces two other forms of a quadratic function. Where ( h, k) is the vertex and x = h is the axis of symmetry. In the previous lesson, it was given that standard form of a quadratic function is SDA NAD Content Standards (2018): AII.5.1, AII.5.3, AII.7.1įigure 1: The cables of the Golden Gate Bridge can be modeled with a quadratic function. Write a quadratic function in intercept form given its x-intercepts.Find the axis symmetry and vertex of a quadratic function.Graph quadratic functions in intercept and general form.
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