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Graphs of Quadratics
Finally, graphs of quadratics. This is another relatively uncommon topic on the test. Back in the Algebra lesson videos, we discussed quadratics. As you may remember, a quadratic expression of x is any expression in which the highest power is x squared. If we set y equal to a quadratic expression of x, this would be a quadratic equation.
So, for example, this is a quadratic equation, this is a quadratic equation. This is a quadratic equation. All of these have a highest power of x squared. Now we can ask, what shape will we get if we graph one of these equations in the x-y plane? Very interesting.
Well, as you may remember from high school, the graph of a parabola of a quadratic equation in the x-y plane is a shape called a parabola. A parabola is a very particular kind of U-shaped curve such as this. And I’ll just point out, not everything U-shaped is a parabola. A parabola has a very specific curvature, just as a circle is, not everything round is a circle, a circle is a very specific thing, a parabola is also a very specific thing.
I’ll also just mention, this is part of the magic of the x-y plane, this connection that we establish between equations and shapes, that is, between algebra and geometry. Parabolas can open up or open down, and can range from narrow to wide. So on the left, we have a narrow parabola opening up, on the right, we have a wide parabola opening down.
Parabolas all have a vertical line of symmetry. What this means is, if we start with any point on the parabola and reflect it over this line, the reflected image will be another point on the parabola. So the parabola, the purple parabola is absolutely symmetrical, it is sym, it is symmetrical over that green line, the line of symmetry. The line of symmetry always passes through the vertex.
Now, on an upward facing parabola, the vertex is the lowest point. On a downward facing parabola, the, the vertex is the highest point. And so, where the, the line of symmetry intersects the parabola, is always the vertex. And if someone tells us the vertex, automatically, they’re telling us the line of symmetry, as well.
Here’s a practice problem, pause the video, and then we’ll talk about this. Okay. So we’re told the vertex so that automatically means that we know the line of symmetry. So the line of symmetry would be the line x equals 2, it would go right up the middle of this parabola.
We have a y-intercept of 9. Well let’s think about this. That y-intercept is two units to the left of the line of symmetry. Well, where would the other point be? It would be two units to the right of the line of symmetry. So we’ll have a y-coordinate of 9 and an x coordinate of 4, in other words, 4, 9.
Sometimes the test will simply give you the graph of the parabola, sometimes it will give you the quadratic equation, and expect you to figure out something about the graph. As with lines, every point on the parabola must satisfy the equation of the parabola. That’s actually a deep idea of the x-y plane. Let’s take a look at a particular equation.
Suppose we have this equation, and we wanna find points on this equation. Well a very easy way would just be to plug in some values. So for example, we plug in x equals 0, we get y equals 4. We plug in x equals 1, we get y equals 3, plug in 2, we get 0, plug in 3, we get negative 5. Less, we should also go in the other direction.
If we plug in negative 1, we get 3, we plug in negative 2, we get 0, if we plug in negative 3, we get negative 5. So we certainly see some symmetry emerging here. And in fact, if we plot these points, then what we’ll get is a graph. And it will be very clear that 0, 4 is the vertex and the line of symmetry is the y-axis.
Standard form for a quadratic equation is y equals a squared plus bx plus c. When a is greater than 0, the parabola opens upward and when a is less than 0 the parabola opens downward. So we can tell whether it’s an upward or downward opening parabola purely by the sign of the quadratic coefficient, the coefficient of x squared.
When the absolute value of a is greater than 1, the parabola is skinny, and when the absolute value of a is less than 1, the parabola is wide. So this gives a way to look at a and tell relatively how skinny or wide the parabola is. When y equals 0, of course those are the intercepts.
These are the solutions of the quadratic equation 0 equals ax squared plus bx plus c. Recall that a quadratic equation can have two, or one, or no solutions. And graphically, this makes sense, we could have a parabola that intersects the x-axis twice, or once, or not at all. Here’s a practice problem, pause the video and then we’ll talk about this.
A parabola has an x-intercept at negative 4, 0. If the vertex is at 2, 5, find the other x-intercept. Well this is interesting because we know the vertex, we know the line of symmetry, the line of symmetry has to be x equals negative 2. Well let’s think about this, the point negative 4, 0 is six units to the left of the line of symmetry.
So its reflection would have to be six units to the right of liners, of line, the line of symmetry, and so that would be the point 8, 0, and that’s the other x-intercept. So notice that all we had to do was think about symmetry, we didn’t have to do any elaborate calculations. Here’s the picture, just to verify what’s going on.
The graph of a quadratic is a parabola. If a is the quadratic coefficient, the multiplier of x squared, then if a is positive that means that the parabola opens upward, when a is negative it means it opens downward. If it has an absolute value greater than 1, then the parabola, parabola is skinny.
If the absolute value of a is less than 1, then the parabola is wide. The x-intercepts of the graph are the solutions of the quadratic set equal to zero. And those are the big ideas of quadratics.
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