41. Compare the parabola defined by each equation with the standard parabola defined by the equation y = x². Describe the corresponding transformations, and include the position of the vertex and the equation of the axis of symmetry.
y = 3x² - 8 (4 marks)
B. y = (x – 6)² + 4 (4 marks)
C. y = -4(x + 3)² - 7 (5 marks)
y = 3x² - 8 (4 marks)
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well that 3 makes it climb up three times as fast
now your original y = x^2 had a vertex at (0,0)
where is it for this one?
for every x, y is 8 less, so it is 8 units lower
vertex at (0,-8), symmetric about that vertical line through x = 0
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B. y = (x – 6)² + 4 (4 marks)
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y - 4 = (x-6)^2
well this time it will go up like the original but x = 6, y = 4, symmetric about x = 6 (note if x = 7, (7-6)^2 = 1 and if x = 5, (5-6)^2 = 1 :). The bottom will be at y = 4 (6,4)
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C. y = -4(x + 3)² - 7 (5 marks)
y+7 = -4 (x+3)^2
this opens down (sheds water) beause y goes to -oo when x gets big + or -
this tops when y = -7 and x = -3
it is 4 times as steep as the original
Thank you so much, I really appreciate it.
You are welcome.
To compare the given parabolas with the standard parabola y = x², we need to analyze the equations and determine the corresponding transformations.
For the equation y = 3x² - 8, let's break it down:
1. Vertex: The vertex of a parabola in the form y = ax² + bx + c is given by the coordinates (-b/2a, f(-b/2a)). Here, a = 3, b = 0, and c = -8. Thus, the x-coordinate of the vertex is -b/2a = 0/(2*3) = 0, and the y-coordinate is f(0) = 3(0)² - 8 = -8. So, the vertex is (0, -8).
2. Axis of Symmetry: The equation of the axis of symmetry is given by x = -b/2a. In this case, it is x = 0.
The transformation for this parabola involves a vertical shift downward by 8 units from the standard parabola. The vertex is located at the origin (0, -8), and the axis of symmetry is the y-axis.
For the equation y = (x - 6)² + 4, let's analyze it as well:
1. Vertex: In this equation, the vertex form is given by (h, k), where h and k are the coordinates of the vertex. Here, h = 6 and k = 4. Therefore, the vertex is (6, 4).
2. Axis of Symmetry: The equation of the axis of symmetry is x = h. In this case, it is x = 6.
The transformation for this parabola involves a horizontal shift to the right by 6 units and a vertical shift upward by 4 units from the standard parabola. The vertex is located at (6, 4), and the axis of symmetry is a vertical line at x = 6.
Finally, let's analyze the equation y = -4(x + 3)² - 7:
1. Vertex: The vertex form of a parabola gives the vertex coordinates as (h, k). In this case, h = -3 and k = -7. Therefore, the vertex is located at (-3, -7).
2. Axis of Symmetry: The equation of the axis of symmetry is x = h. Here, it is x = -3.
The transformation for this parabola includes a horizontal shift to the left by 3 units, a vertical reflection, a vertical stretch by a factor of 4, and a vertical shift downward by 7 units from the standard parabola. The vertex is positioned at (-3, -7), and the axis of symmetry is a vertical line at x = -3.
To summarize:
- The parabola defined by y = 3x² - 8 has a vertex at (0, -8) and an axis of symmetry along the y-axis.
- The parabola defined by y = (x - 6)² + 4 has a vertex at (6, 4) and an axis of symmetry at x = 6.
- The parabola defined by y = -4(x + 3)² - 7 has a vertex at (-3, -7) and an axis of symmetry at x = -3.