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I want the steps and explain it, thanks you.
Ans: (a) 9
(b) (0, -7)
(c) 14
Date Posted: 3 years ago
Views: 288
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i)
P is the vertex of the graph, which means it is the highest point of the graph, whereby the value of y is the maximum at P.
Since the question says the maximum value of the function is 9, y = 9 at P. P is what we call the maximum point.
y = -(x - 4)² + k
For the maximum point, we want the max value of y. This means that the negative value in the equation above, -(x-4)² should be as small as possible. k is a constant so we don't have to worry about it.
This is achieved when x = 4, which turn means that y = -(4-4)² + k = -0² + k = k
At other values of x, you will always get a negative value for -(x-4)², because (x - 4)² itself will be always positive (the square of any real non-zero value is always > 0) and the negative sign outside makes it negative.
Eg. When x = 0, y = -(0 - 4)² + k = -(-4)² + k = -16 + k = k - 16
When x = 7, y = -(7 - 4)² + k = -(3)² + k = -9 + k = k - 9
When x = -2, y = -(-2 - 4)² + k = -(-6)² + k = -36 + k = k - 36
These are all smaller than k (when x = 4)
Therefore, since y = 9, x = 4 at the maximum point,
9 = -(4-4)² + k
9 = 0 + k
k = 9
y = a(x-h)² + k is known as the vertex form of a parabola, whereby:
①a is the coefficient of x² (negative means the graph is downward sloping, n-shaped and positive means the graph is upward sloping, u-shaped)
② (h,k) is the coordinates of the vertex. In this question, h = 4, k = 9, a = -1.
For further reading :
https://www.purplemath.com/modules/sqrvertx.htm
P is the vertex of the graph, which means it is the highest point of the graph, whereby the value of y is the maximum at P.
Since the question says the maximum value of the function is 9, y = 9 at P. P is what we call the maximum point.
y = -(x - 4)² + k
For the maximum point, we want the max value of y. This means that the negative value in the equation above, -(x-4)² should be as small as possible. k is a constant so we don't have to worry about it.
This is achieved when x = 4, which turn means that y = -(4-4)² + k = -0² + k = k
At other values of x, you will always get a negative value for -(x-4)², because (x - 4)² itself will be always positive (the square of any real non-zero value is always > 0) and the negative sign outside makes it negative.
Eg. When x = 0, y = -(0 - 4)² + k = -(-4)² + k = -16 + k = k - 16
When x = 7, y = -(7 - 4)² + k = -(3)² + k = -9 + k = k - 9
When x = -2, y = -(-2 - 4)² + k = -(-6)² + k = -36 + k = k - 36
These are all smaller than k (when x = 4)
Therefore, since y = 9, x = 4 at the maximum point,
9 = -(4-4)² + k
9 = 0 + k
k = 9
y = a(x-h)² + k is known as the vertex form of a parabola, whereby:
①a is the coefficient of x² (negative means the graph is downward sloping, n-shaped and positive means the graph is upward sloping, u-shaped)
② (h,k) is the coordinates of the vertex. In this question, h = 4, k = 9, a = -1.
For further reading :
https://www.purplemath.com/modules/sqrvertx.htm
b)
The graph cuts the y-axis at Q. At any point on the y-axis, x = 0. So at Q, the x-coordinate of the graph is also 0.
Since we now know the equation of the graph to be y = -(x - 4)² + 9, substitute x = 0 to find the y-coordinate at Q.
y = -(0 - 4)² + 9
y = -(4²) + 9
y = -16 + 9 = -7
So the coordinates of Q are (0,-7)
The graph cuts the y-axis at Q. At any point on the y-axis, x = 0. So at Q, the x-coordinate of the graph is also 0.
Since we now know the equation of the graph to be y = -(x - 4)² + 9, substitute x = 0 to find the y-coordinate at Q.
y = -(0 - 4)² + 9
y = -(4²) + 9
y = -16 + 9 = -7
So the coordinates of Q are (0,-7)
Slight error in the last few steps. It should be
y = -(-4)² + 9 instead of y = -(4²) + 9
y = -(-4)² + 9 instead of y = -(4²) + 9
c)
This one is easy. We just need the base and height of the triangle.
For the base, we can use the length of OQ. Since OQ is on the y-axis, it is a vertical line.
The length of OQ is simply the absolute difference of the y-coordinates of the origin O (coordinates(0,0)) and Q.
Length of OQ = (0 - (-7)) units = (0 + 7) units = 7 units
For the height, we'll use the length of the perpendicular from P to the y-axis.
Now P has the coordinates (4,9). The 4 means that it is 4 units horizontally right of the y-axis. The 9 means that it is 9 units vertically/directly above the x-axis.
This means that the length of the perpendicular is 4 units, since the perpendicular is effectively the same as this horizontal distance.
So the triangle has height of 4 units, base of 7 units.
Area of triangle = ½ × base × height = ½ × 7 units × 4 units
= 14 units ²
This one is easy. We just need the base and height of the triangle.
For the base, we can use the length of OQ. Since OQ is on the y-axis, it is a vertical line.
The length of OQ is simply the absolute difference of the y-coordinates of the origin O (coordinates(0,0)) and Q.
Length of OQ = (0 - (-7)) units = (0 + 7) units = 7 units
For the height, we'll use the length of the perpendicular from P to the y-axis.
Now P has the coordinates (4,9). The 4 means that it is 4 units horizontally right of the y-axis. The 9 means that it is 9 units vertically/directly above the x-axis.
This means that the length of the perpendicular is 4 units, since the perpendicular is effectively the same as this horizontal distance.
So the triangle has height of 4 units, base of 7 units.
Area of triangle = ½ × base × height = ½ × 7 units × 4 units
= 14 units ²
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