Jack threw a rock into the lake and is represented by the equation f(x)=-3x^2-6x+21, where f(x) is the height in feet and x, is the time in seconds. Find the h, of the vertex (h,k) and find the zeros. What does each mean in the context of the question?
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Answer:Given equation that shows the height in feet after x seconds,[tex]f(x) = -3x^2 - 6x + 21[/tex][tex]f(x) = -3(x^2 + 2x) + 21[/tex][tex]f(x) = -3(x^2 + 2x + 1) + 21 + 3[/tex][tex]f(x) = -3(x+1)^2+24[/tex]Since, the vertex form of a quadratic equation is,[tex]f(x) = a(x-h)^2 + k[/tex]Where,(h, k) is the vertex of the equation.By comparing,Vertex = (h, k) = (-1, 24)i.e. h = -1Now, f(0) = 21⇒ The initial height of the rock from the lake is 21 ft,Now, for zeroes,f(x) = 0,[tex]-3x^2 - 6x + 21=0[/tex]By quadratic formula,[tex]x = \frac{6 \pm \sqrt{(-6)^2 - 4\times -3\times 21}}{-6}[/tex][tex]=\frac{6\pm \sqrt{36 + 252}}{-6}[/tex][tex]\implies x = 1.828\text{ or }x = -3.828[/tex]Since, number of seconds can not be zero,∴ x = 1.828 secondsi.e. the rock will reach in the lack after 1.828 seconds.