Published at : 27 Dec 2020

Quickly master how to find characteristics of quadratic functions. Watch more lessons like this and try our practice at https://www.studypug.com/algebra-help/quadratic-functions/characteristics-of-quadratic-functions

Three properties that are universal to all quadratic functions: 1) The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior); 2) The domain of a quadratic function is all real numbers; and 3) The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.

Let's learn some vocabulary for quadratic functions. First, vertex. What is a vertex of a quadratic function? Well, the word vertex means peak. So the vertex of a quadratic function is the peak point on a parabola. Which means for a parabola opening down, the vertex is the highest point. Now for a parabola opening up, the vertex is the lowest point. So in this case, we have a parabola opening up, so the vertex is the lowest point, right here. Sitting at x equals 1, y equals negative 9. So the vertex is at 1 and negative 9. Next, 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 is simply x equals 1. Because for every point on this vertical line, the x coordinate is always 1. So in this case, the equation of the axis of symmetry is x equals 1. 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, 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 this point x value is positive 5. So a parabola is defined for all positive integers for x. 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. Parabola can take on any positive numbers for x. Now, guess what? Parabola can also take on any 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. 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 also take 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 write a mathematical notation like this, x is all real numbers.

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Three properties that are universal to all quadratic functions: 1) The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior); 2) The domain of a quadratic function is all real numbers; and 3) The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.

Let's learn some vocabulary for quadratic functions. First, vertex. What is a vertex of a quadratic function? Well, the word vertex means peak. So the vertex of a quadratic function is the peak point on a parabola. Which means for a parabola opening down, the vertex is the highest point. Now for a parabola opening up, the vertex is the lowest point. So in this case, we have a parabola opening up, so the vertex is the lowest point, right here. Sitting at x equals 1, y equals negative 9. So the vertex is at 1 and negative 9. Next, 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 is simply x equals 1. Because for every point on this vertical line, the x coordinate is always 1. So in this case, the equation of the axis of symmetry is x equals 1. 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, 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 this point x value is positive 5. So a parabola is defined for all positive integers for x. 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. Parabola can take on any positive numbers for x. Now, guess what? Parabola can also take on any 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. 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 also take 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 write a mathematical notation like this, x is all real numbers.

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