The exact value is found by making use of order of operations. The
functions can be resolved using the characteristics of quadratic functions.
Correct responses:
- x = -4, y = 12
- When P = 1, V = 6
ii. The function has a minimum point
iii. The value of <em>x</em> at the minimum point, is <u>1.25</u>
iv. The equation of the axis of symmetry is <u>x = 1.25</u>
<h3>Methods by which the above responses are found</h3>First part:
The given expression, , can be simplified using the algorithm for arithmetic operations as follows;
Second part:
y = 8 - x
2·x² + x·y = -16
Therefore;
2·x² + x·(8 - x) = -16
2·x² + 8·x - x² + 16 = 0
x² + 8·x + 16 = 0
(x + 4)·(x + 4) = 0
- <u>x = -4</u>
y = 8 - (-4) = 12
- <u>y = 12</u>
Third part:
(i) P varies inversely as the square of <em>V</em>
Therefore;
V = 3, when P = 4
Therefore;
K = 3² × 4 = 36
When P = 1, we have;
- When P = 1, V =<u> 6</u>
Fourth Part:
Required:
Solving for <em>x</em> in the equation; 2·x² + 5·x - 3 = 0
Solution:
The equation can be simplified by rewriting the equation as follows;
2·x² + 5·x - 3 = 2·x² + 6·x - x - 3 = 0
2·x·(x + 3) - (x + 3) = 0
(x + 3)·(2·x - 1) = 0
Fifth part:
The given function is; f(x) = 2·x² - 5·x + 8
i. Required; To write the function in the form a·(x + b)² + c
The vertex form of a quadratic equation is f(x) = a·(x - h)² + k, which is similar to the required form
Where;
(h, k) = The coordinate of the vertex
Therefore, the coordinates of the vertex of the quadratic equation is (b, c)
The x-coordinate of the vertex of a quadratic equation f(x) = a·x² + b·x + c, is given as follows;
Therefore, for the given equation, we have;
Therefore, at the vertex, we have;
a = The leading coefficient = 2
b = -h
c = k
Which gives;
Therefore;
ii. The coefficient of the quadratic function is <em>2</em> which is positive, therefore;
- <u>The function has a minimum point</u>.
iii. The value of <em>x</em> for which the minimum value occurs is -b = h which is therefore;
- The x-coordinate of the vertex = h = -b =<u> 1.25 </u>
iv. The axis of symmetry is the vertical line that passes through the vertex.
Therefore;
- The axis of symmetry is the line <u>x = 1.25</u>.
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