Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-4 x^{2}+32 x-20 $$

Short answer

The vertex of the function \(y=-4x^2+32x-20\) is at the point (4, 4). The parabola opens downward.

Step-by-step solution

Identify the coefficients

The coefficients from the function are: \(a = -4\), \(b = 32\), and \(c = -20\).

Calculate the vertex

Use the formula to find the x-coordinate of the vertex \(x = -b/2a\), so \(x = -32/(2*(-4)) = 4\). To find the y-coordinate of the vertex, substitute the x-value into the function: \(y = -4*4^2 + 32*4 - 20 = 4\). Thus, the coordinates of the vertex are (4, 4).

Plot the vertex and draw the graph

Plot the vertex at the point (4, 4). As the coefficient of \(x^2\) is negative, the parabola opens downward. At \(x = 0\), \(y = -20\). And at \(x = 8\), the function also equals \(y = -20\). So plot these points and then sketch the graph by connecting the points. The graph is symmetric with respect to the line \(x = 4\).