Questions: Find the coordinates of the vertex for the parabola defined by the given f(x)=4x^2-8x+1 The vertex is . (Type an ordered pair.)

Transcript text: Find the coordinates of the vertex for the parabola defined by the gi \[ f(x)=4 x^{2}-8 x+1 \] The vertex is $\square$ . (Type an ordered pair.)

Solution

Solution Steps

Solution

To find the vertex of a parabola given by the equation $f(x) = ax^2 + bx + c$, we use the formulas:

$h = -\frac{b}{2a}$ to find the x-coordinate of the vertex.
$k = f(h) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$ to find the y-coordinate of the vertex.

Step 1: Calculate the x-coordinate of the vertex ($h$)

Substitute $a = 4$ and $b = -8$ into $h = -\frac{b}{2a}$ to get $h = -\frac{-8}{2*4} = 1$.

Step 2: Calculate the y-coordinate of the vertex ($k$)

Substitute $h = 1$, $a = 4$, $b = -8$, and $c = 1$ into $k = a(h)^2 + bh + c$ to get $k = 4_(1)^2 - 8_1 + 1 = -3$.