Quadratic equation and quadratic functions mathquestion.in

 Sitting at x equals 1, y equals negative 9. So the vertex is at 1 and negative 9. Next, the axis of symmetry. The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two equal halves, like this. And the axis of symmetry must always pass through the vertex. The symmetry behavior can be best understood when we fold the parabola in half along the axis of symmetry. Then the right half of the parabola will completely overlap the left half of the parabola. Can you see that? Let me show you. Here we have a parabola, just like this one, and the axis of symmetry runs down the middle of a the parabola.

 Now, I'm going to fold the graph along the axis of symmetry.Can you see that? The two halves of the parabola completely overlap each other. Yeah? Cool. Now let's determine the equation of the axis of symmetry. For a vertical line that passes through x equals 1, the equation of the line issimply x equals 1. Because for every point on this vertical line, the coordinate is always 1. So in this case, the equation of the axis of symmetry is equaled 1 quadratic functions

Quadratic  equation . Next, y-intercept. The y-intercept is the point where the parabola intersects the y-axis. So in this case, that's right here and the y-intercept is negative 8. Next, x-intercepts. The x-intercepts are the points where the parabola intersects the x-axis. So in this case, the parabola intersects the x axis at two points. One at x equals negative 2 another one at x equals 4.Next, quadratic functions domain. By definition, domain is the set of x values for which a function is defined. So as we can see, a parabola is defined for all x values. Let me show you.This point, the x value is positive 1. This point x value is positive 2. This point x value is positive 3. This point x value is positive 4, 

and thispoint x value is positive 5. So a parabola is defined for all positive integers forx. Now if you take any points in between, now see, these points the x values are positive decimal numbers, right? Yeah. So a parabola can take on any positive values for x, whether they are integers or whether they are decimal numbers. Parabolacan take on any positive numbers for x. Now, guess what? Parabola can also take onany negative values for x. For example, this point x value is negative 1.This point x value is negative 2. This point x value is negative 3. You see, negative integers. quadratic functions

Quadratic  equation  Now, for any point in between them, for any points between the negative integers, this point the x values are all negative decimal numbers.So in conclusion, parabola can take on any positive values for x. Parabola can alsotake on any negative values for x and parabola can even take on zero for x.That would be this point, for the y intercept, the x value is zero, right?So therefore, the domain for a quadratic function is all real numbers. And we writea mathematical notation like this

Quadratic equation , x is all real numbers. This symbol means "is"and this symbol here, represents, see this symbol here is an R with an extra bar, represents "all real numbers." In fact, the domain for any quadratic function is always x is all real numbers. Next, range. quadratic functions By definition, range is theset of y values for which a function is defined. So here, the parabola only goesas low as right here, where the y value is negative 9. However, you will go ashigh it likes and therefore we say the range . . . Range talks about y right? Sothe range is y must be negative 9 and anything above.

 So the range is y is greater than or equal to negative 9. Does this parabola have a minimum value ora maximum value? Well first of all, in math, whenever we talk about the value of a function, we are always referring to the y value. Remember that. The value of function always refers to they value. Now, since the parabola can go as high asit likes, there's no maximum value. However, a parabola can only go as low asright here, where the y value is negative 9. Therefore, we say this parabola has a minimum value of negative 9.