91Ó°ÊÓ

Write down the quadratic equation we need to work with: \[ y = \frac{1}{2}x^2 + 2x - 6 \]

Select a range of x-values to calculate corresponding y-values. A good range to start with might be -3 to 3. This allows us to see the behavior of the quadratic equation across positive, negative, and zero x-values.

Substitute each chosen x-value into the equation and solve for y. For each x, compute:1. For \( x = -3 \), \[ y = \frac{1}{2}(-3)^2 + 2(-3) - 6 \] \[ y = \frac{1}{2}(9) - 6 - 6 \] \[ y = 4.5 - 6 - 6 = -7.5 \]2. For \( x = -2 \), \[ y = \frac{1}{2}(-2)^2 + 2(-2) - 6 \] \[ y = \frac{1}{2}(4) - 4 - 6 \] \[ y = 2 - 4 - 6 = -8 \]3. For \( x = -1 \), \[ y = \frac{1}{2}(-1)^2 + 2(-1) - 6 \] \[ y = \frac{1}{2}(1) - 2 - 6 \] \[ y = 0.5 - 2 - 6 = -7.5 \]4. For \( x = 0 \), \[ y = \frac{1}{2}(0)^2 + 2(0) - 6 \] \[ y = 0 - 6 = -6 \]5. For \( x = 1 \), \[ y = \frac{1}{2}(1)^2 + 2(1) - 6 \] \[ y = \frac{1}{2}(1) + 2 - 6 \] \[ y = 0.5 + 2 - 6 = -3.5 \]6. For \( x = 2 \), \[ y = \frac{1}{2}(2)^2 + 2(2) - 6 \] \[ y = 2 + 4 - 6 = 0 \]7. For \( x = 3 \), \[ y = \frac{1}{2}(3)^2 + 2(3) - 6 \] \[ y = 4.5 + 6 - 6 \] \[ y = 4.5 \]

Take the calculated pairs of (x, y) values and plot them on a graph. The points to plot are: - (-3, -7.5) - (-2, -8) - (-1, -7.5) - (0, -6) - (1, -3.5) - (2, 0) - (3, 4.5)

Once the points from Step 4 are plotted, connect them with a smooth curve to reveal the parabolic shape of the quadratic function. Ensure the curve reflects the function's symmetry and opens upwards.