The parabola shown is the graph of f(x) = Ax^{2} + 2x + C. The x-intercepts of the graph are at -4 and -3. Find the exact value of the y-intercept and the coordinates of the vertex of the graph (expressed in terms of rational numbers and radicals).

Tolnaio 2020-12-06 Answered
The parabola shown is the graph of f(x)=Ax2 + 2x + C.
The x-intercepts of the graph are at 4 and 3. Find the exact value of the y-intercept and the coordinates of the vertex of the graph (expressed in terms of rational numbers and radicals).
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Expert Answer

jlo2niT
Answered 2020-12-07 Author has 96 answers
Given:
f(x)=Ax2 + 2x + C
x intercept are 4 and 3
We need to write x intercept in factor form
If p and q are the factors then intercepts form of quadratic equation is
f(x)=a(x  p)(x  q)
4 and 3 are x intercepts
f(x)=a(x  (4))(x  (3))
f(x)=a(x + 4)(x + 3)
f(x)=a(x2 + 7x + 12)
f(x)=ax2 + 7ax + 12a
Now compare f(x) with out given f(x)
f(x)=Ax2 + 2x + C
f(x)=ax2 + 7ax + 12a
7a=2
a=27
Replace the value in f(x), then f(x) becomes
f(x)=ax2 + 7ax + 12a
f(x)=27x2 + 7(27)x + 12(27)
f(x)=27x2 + 2x + 247
To find out vertex we use formula
x=b2a
b=2, a=27
x=22(27)=247=2  74=72
now we find out y
f(x)=27x2 + 2x + 247
x=72
f(x)=27(72)2 + 2(72) + 247=114
Vertex
(72  114)
Now find out y intercept when x=0
f(x)=27x2 + 2x + 247
f(0)=27(0)2 + 2(0) + 247=247
y intercept is (247)
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