The equation of the axis of symmetry is [tex]\mathbf{x =1}[/tex], the value of a is 2, and the value of c is 1.
The function is given as:
[tex]\mathbf{f(x) = ax^2 -4x - c}[/tex]
Line L is represented as:
[tex]\mathbf{(x,y) = (-1,5)(3,5)}[/tex]
Substitute the above values in: [tex]\mathbf{f(x) = ax^2 -4x - c}[/tex]
So, we have:
[tex]\mathbf{5 = a(-1)^2 - 4(-1) - c}[/tex]
[tex]\mathbf{5 = a(3)^2 - 4(3) - c}[/tex]
Expand [tex]\mathbf{5 = a(-1)^2 - 4(-1) - c}[/tex]
[tex]\mathbf{5 = a + 4 - c}[/tex]
Expand [tex]\mathbf{5 = a(3)^2 - 4(3) - c}[/tex]
[tex]\mathbf{5 = 9a - 12 - c}[/tex]
Subtract [tex]\mathbf{5 = a + 4 - c}[/tex] and [tex]\mathbf{5 = 9a - 12 - c}[/tex]
[tex]\mathbf{5 - 5 = a - 9a + 4 + 12 -c + c}[/tex]
[tex]\mathbf{0 = - 8a + 16}[/tex]
Collect like terms
[tex]\mathbf{8a = 16}[/tex]
Divide both sides by 8
[tex]\mathbf{a = 2}[/tex]
Substitute [tex]\mathbf{a = 2}[/tex] in [tex]\mathbf{5 = a + 4 - c}[/tex]
[tex]\mathbf{5 = 2 + 4 - c}[/tex]
[tex]\mathbf{5 = 6 - c}[/tex]
Subtract 6 from both sides
[tex]\mathbf{-1 = - c}[/tex]
So, we have:
[tex]\mathbf{c = 1}[/tex]
Recall that:
[tex]\mathbf{f(x) = ax^2 -4x - c}[/tex]
Where:
[tex]\mathbf{a = 2}[/tex] and [tex]\mathbf{b = -4}[/tex]
The equation of the axis of symmetry is:
[tex]\mathbf{x = -\frac{b}{2a}}[/tex]
So, we have:
[tex]\mathbf{x = -\frac{-4}{2 \times 2}}[/tex]
[tex]\mathbf{x = -\frac{-4}{4}}[/tex]
[tex]\mathbf{x =1}[/tex]
Hence, the equation of the axis of symmetry is [tex]\mathbf{x =1}[/tex], the value of a is 2, and the value of c is 1.
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