Answer:
D) c = 4
C) b = -4 or b = 4
Step-by-step explanation:
The minimum/maximum value of a quadratic function is the y-coordinate of the function's vertex.
The x-coordinate of a function's vertex can be found using the following formula:
[tex]\textsf{$x$-coordinate of the vertex}=-\dfrac{b}{2a}\quad \textsf{(for $ax^2+bx+c$)}.[/tex]
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Given function:
[tex]f (x) = 4x^2+ 12x + c[/tex]
Therefore,
[tex]\textsf{$x$-coordinate of the vertex}=-\dfrac{b}{2a}=-\dfrac{12}{2(4)}=-1.5[/tex]
Substitute the found x-value of the vertex into the function, equate it to -5, then solve for c:
[tex]\begin{aligned} f(-1.5)&=-5\\\implies4(-1.5)^2+12(-1.5)+c&=-5\\4(2.25)-18+c&=-5\\9-18+c&=-5\\-9+c&=-5\\c&=4\end{aligned}[/tex]
Therefore, c = 4.
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Given function:
[tex]f (x) = -x^2 + bx + 7[/tex]
Therefore,
[tex]\textsf{$x$-coordinate of the vertex}=-\dfrac{b}{2(-1)}=0.5b[/tex]
Substitute the found x-value of the vertex into the function, equate it to 11, then solve for b:
[tex]\begin{aligned} f(0.5b)&=11\\\implies-(0.5b)^2+b(0.5b)+7&=11\\-0.25b^2+0.5b^2+7&=11\\0.25b^2&=4\\b^2&=16\\\sqrt{b^2}&=\sqrt{16}\\b&=\pm4\end{aligned}[/tex]
Therefore, b = -4 or b = 4.
