Questions: The vertex of the parabola defined by f(x)=-5x^2+4x+1 is the point ( , ).

Transcript text: The vertex of the parabola defined by \[ f(x)=-5 x^{2}+4 x+1 \] is the point $\square$ , $\square$ ).

Solution

Solution Steps

To find the vertex of a parabola given by the quadratic function $ f(x) = ax^2 + bx + c $, we use the vertex formula. The x-coordinate of the vertex is given by $ x = -\frac{b}{2a} $. Once we have the x-coordinate, we substitute it back into the function to find the y-coordinate.

Step 1: Find the x-coordinate of the vertex

To find the x-coordinate of the vertex of the parabola defined by the function $ f(x) = -5x^2 + 4x + 1 $, we use the formula: \[ x = -\frac{b}{2a} \] Substituting $ a = -5 $ and $ b = 4 $: \[ x = -\frac{4}{2 \cdot -5} = 0.4 \]

Step 2: Find the y-coordinate of the vertex

Next, we substitute the x-coordinate back into the function to find the y-coordinate: \[ f(0.4) = -5(0.4)^2 + 4(0.4) + 1 \] Calculating this gives: \[ f(0.4) = -5(0.16) + 1.6 + 1 = -0.8 + 1.6 + 1 = 1.8 \]

Step 3: State the vertex

Thus, the vertex of the parabola is the point: \[ (0.4, 1.8) \]