Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-\frac{1}{3} x^{2}+2 x-3 $$

Short answer

The graph is a parabola that opens downwards, with its vertex at the point (3, -2).

Step-by-step solution

Convert the given function into vertex form

To convert into the vertex form, use the formula \( h = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Then substitute \( h \) into the standard function to find \( k \), the y-coordinate of the vertex. Here \( a = -\frac{1}{3} \), \( b = 2 \) and \( c = -3 \). We get \( h = -\frac{2}{2*(-1/3)} = 3 \). Substituting \( x = h \) in the function, we get \( y = -\frac{1}{3}*3^{2}+2*3-3 = -2 \). Therefore, the vertex is at (3, -2).

Identify the direction of the parabola

The direction of the parabola depends on the coefficient of \( x^{2} \) in the quadratic function. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards. Here \( a = -\frac{1}{3} \), so the parabola opens downwards

Plot the vertex and draw the parabola

Plot the vertex at the coordinate (3,-2) on graph paper. Since the parabola opens downwards, draw a parabola opening downwards from the vertex. The sketch should be a rough estimate of the curve's shape. The more points plotted, the more accurate the graph.