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:
[tex]\displaystyle i. \hspace{0.1 cm} \underline{ f(x) = 2 \cdot \left(x - 1.25 \right)^2 + 4.875 }[/tex]
ii. The function has a minimum point
iii. The value of x at the minimum point, is 1.25
iv. The equation of the axis of symmetry is x = 1.25
First part:
The given expression, [tex]\displaystyle \mathbf{ 1\frac{4}{7} \div \frac{2}{3} -1\frac{5}{7}}[/tex], 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
y = 8 - (-4) = 12
Third part:
(i) P varies inversely as the square of V
Therefore;
[tex]\displaystyle P \propto \mathbf{\frac{1}{V^2}}[/tex]
[tex]\displaystyle P = \frac{K}{V^2}[/tex]
V = 3, when P = 4
Therefore;
[tex]\displaystyle 4 = \frac{K}{3^2}[/tex]
K = 3² × 4 = 36
[tex]\displaystyle V = \sqrt{\frac{K}{P}[/tex]
When P = 1, we have;
[tex]\displaystyle V =\sqrt{ \frac{36}{1} } = 6[/tex]
Fourth Part:
Required:
Solving for x 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;
[tex]\displaystyle h = \mathbf{ \frac{-b}{2 \cdot a}}[/tex]
Therefore, for the given equation, we have;
[tex]\displaystyle h = \frac{-(-5)}{2 \times 2} = \mathbf{ \frac{5}{4}} = 1.25[/tex]
Therefore, at the vertex, we have;
[tex]k = \displaystyle f\left(1.25\right) = 2 \times \left(1.25\right)^2 - 5 \times 1.25 + 8 = \frac{39}{8} = 4.875[/tex]
a = The leading coefficient = 2
b = -h
c = k
Which gives;
[tex]\displaystyle f(x) \ in \ the \ form \ a \cdot (x + b)^2 + c \ is \ f(x) = 2 \cdot \left(x + \left(-1.25 \right) \right)^2 +4.875[/tex]
Therefore;
ii. The coefficient of the quadratic function is 2 which is positive, therefore;
iii. The value of x for which the minimum value occurs is -b = h which is therefore;
iv. The axis of symmetry is the vertical line that passes through the vertex.
Therefore;
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