What is the graph of f(x)=3x-x^2 ?

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Answer 1 · 2018-03-11T21:33:00

graph{3x-x^2 [-10, 10, -5, 5]}

The parabola opens downward because of the negative sign in front of the $x^{2}$ $x^2$ .

The $x$ $x$ -intercepts are $(0, 0)$ $(0,0)$ and $(3, 0)$ $(3,0)$ because

$3 x - x^{2} = 0$ $3x-x^2=0$

factored equals

$(x) (x - 3) = 0$ $(x)(x-3)=0$

When each of the parenthetical expressions is set equal to zero, you get

$x = 0$ $x=0" "$ and $x = 3$ $" "x=3$

The vertex is $(\frac{3}{2}, \frac{9}{4})$ $(3/2, 9/4)$ because the $x$ $x$ -coordinate of the vertex is

$x-coord. vertex = - \frac{b}{2 a}$ $"x-coord. vertex" = -b/(2a)$

$= - \frac{3}{2 (- 1)} = \frac{3}{2}$ $= -(3)/(2(-1))=3/2$

Then, you plug $\frac{3}{2}$ $3/2$ into the equation for $x$ $x$ and get

$f (\frac{3}{2}) = 3 (\frac{3}{2}) - {(\frac{3}{2})}^{2} = \frac{9}{4}$ $f(3/2) = 3(3/2)-(3/2)^2 = 9/4$