y=×2-6×-3 transform the quadratic function
Answers
The general shape of a parabola is the shape of a "pointy" letter "u," or a slightly rounded letter, "v." You may encounter a parabola that is "laying on it's side," but we won't discuss such a parabola here because it is not a function as it would not pass the Vertical Line Test.
Parabolas are in one of two forms. The first form is called the standard form, y = ax2 + bx + c. The second form is called the vertex-form or the a-h-k form, y = a(x - h)2 + k.
Parabolas in the standard from y = ax2 + bx + c.
Let's trying graphing another parabola where a = 1, b = -2 and c = 0. So, we would have the equation, y = x2- 2x. Let's substitute the same values in for x as we did in the chart above and see what we get for y.
x
y = x2
y = x2 - 2x
(x, y)
-3
(-3)2-2x
15
(-3, 15)
-2
(-2)2-2x
8
(-2, 8)
-1
(-1)2-2x
3
(-1, 3)
0
(0)2-2x
0
(0, 0)
1
(1)2-2x
-1
(1, -1)
2
(2)2-2x
0
(2, 0)
3
(3)2-2x
3
(3, 3)
Let's graph this function.
Graph of the function y = x2 - 2x
What are the x-and y-intercepts? What is the lowest point on the graph?
Here, we see again that the x- and y-intercepts are both (0, 0), as the parabola crosses through the origin. The lowest point on the graph is (1, -1) and is called the vertex. If you draw a vertical line through the vertex, it will split the parabola in half so that either side of the vertical line is symmetric with respect to the other side.
This vertical line is called the line of symmetry or axis of symmetry. Since the line of symmetry will always be a vertical line in all of our parabolas, the general formula for the line will be x = c.
Remember from earlier lessons that vertical lines are always in the form x = c. To find the equation of the line of symmetry, it will always be y = c, where c is always the x-value of the vertex (x, y). Remember, to graph a vertical line, go across the x-axis to the value of "c" where the equation indicates, x = c, and draw the vertical line. So, in this case, the line of symmetry would be x = 1.
The vertex is the lowest point on the parabola if the parabola opens upward and is the highest point on the parabola if the parabola opens downward.
Now let's try graphing the parabola: y = -3x2 + x + 1. Substitute our standard values in for x and solve for y as illustrated in the chart below:
x
y = x2
y = -3x2+x + 1
(x, y)
-3
-3 (-3)2+ x + 1
-29
(-3, -29)
-2
-3 (-2)2+ x + 1
-13
(-2, -13)
-1
-3 (-1)2+ x + 1
-3
(-1, -3)
0
-3 (0)2+ x + 1
1
(0, 1)
1
-3 (1)2+ x + 1
-1
(1, -1)
2
-3 (2)2+ x + 1
-9
(2, -9)
3
-3 (3)2+ x + 1
-23
(3, -23)
The points and the graph through these points are shown below.
Graph of the quadratic function y = -3x2 + x + 1
What is the y-intercept? Can you estimate the x-intercepts? Can you estimate the vertex? What is the general shape of the parabola?
Remember, to find the y-intercept of any equation, we can always substitute 0 in for x and solve for y. The actual point of the y-intercept is (0, y), so x is always 0.
If substitute 0 in for x, we'll get y = 1 as indicated in the chart above. So our y-intercept is (0, 1). You should be able to also see the y-intercept on the graph.
What about the x-intercepts? There are two in this case, at approximately x = -0.8 and x = 0.4. We'll estimate them now, as we will find out how to calculate them in detail in the next lesson, "The Quadratic Formula."
For now, remember that you would solve for the x-intercepts by substituting 0 in for y and solving for x, as you would for any equation. If we substituted 0 in for y, we would get the equation 0 = -3x2 + x + 1. We would solve for the values of x using the quadratic formula. If you know the quadratic formula, go ahead and solve for the x-intercepts. If you don't know the quadratic formula, not to worry, you're not supposed to! We'll come back to this equation in detail in the next lesson.
What about the vertex? You can't really tell the exact value of the vertex just by looking at the graph. It looks like the x-value of the vertex is a little less than 1/4 of the way from the origin to x = 1, and the y-value of the vertex is a little more than 1. But what is the vertex exactly?
The vertex is an important coordinate to find because we know that the graph of the parabola is symmetric with respect to the vertical line passing through the vertex. The coordinate of the vertex of a quadratic equation in standard form (y = ax2 + bx + c) is (-b/2a, f(-b/2a)), where x = -b/2a and y = f(-b/2a).
This means that to find the x-value of the vertex in the equation, y = -3x2 + x + 1, use the formula that x = -b/2a. In this equation, "b" is the coefficient of the x-term and "a", like always, is the coefficient of the x2