91Ó°ÊÓ

The vertex of a parabola is a key feature to understand. It's the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. For the function in question, \( g(x) = x^{2} + 3x - 10 \), we can calculate the vertex using the formula for the vertex of a quadratic equation in the form \( ax^2 + bx + c \). The \(x\)-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \(a = 1\) and \(b = 3\), so the \( x \)-coordinate of the vertex is \( x = -\frac{3}{2(1)} = -\frac{3}{2} \).

To find the \(y\)-coordinate, substitute \( x = -\frac{3}{2} \) back into the original function:
\[ g\bigg( -\frac{3}{2} \bigg) = \bigg( -\frac{3}{2} \bigg)^2 + 3\bigg( -\frac{3}{2} \bigg) - 10 \] The calculation becomes
\[ g\big( -\frac{3}{2} \big) = \frac{9}{4} - \frac{9}{2} - 10 = \frac{9}{4} - \frac{18}{4} - \frac{40}{4} = -\frac{49}{4} \] Thus, the vertex of the parabola is \( \left( -\frac{3}{2}, -\frac{49}{4} \right) \). This point tells you the parabola's lowest point since the parabola opens upwards.

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror images. It passes through the vertex, making it a crucial part of graphing quadratic functions. For the quadratic function \( g(x) = x^2 + 3x - 10 \) , the axis of symmetry can be calculated using the formula \( x = -\frac{b}{2a} \). Here, with \( a = 1 \) and \( b = 3 \), the axis of symmetry is
\[ x = -\frac{3}{2(1)} = -\frac{3}{2} \] This vertical line helps in plotting other points on the graph because it means that for every point \( (x, y) \) on one side of the axis, there is a corresponding point \( (-x, y) \) on the opposite side.

For easier reference:

• The equation of the axis of symmetry for this example is \( x = -\frac{3}{2} \).
• It divides the parabola into two symmetrical halves.

Plotting points on the graph helps to visualize the shape and position of the parabola. Start by plotting the vertex which for the function \( g(x) = x^2 + 3x - 10 \) is at \( \left( -\frac{3}{2}, -\frac{49}{4} \right) \).

• First, draw the vertical axis of symmetry at \( x = -\frac{3}{2} \).
• Next, use the axis of symmetry to help identify other key points on the graph. Pick points to the left and right of \( x = -\frac{3}{2} \) and then find their corresponding \( y \)-values.

Let's choose a couple of points:

• For \( x = -2 \):
  Use \( g(-2) = (-2)^2 + 3(-2) - 10 = 4 - 6 - 10 = -12 \)
• For \( x = -1 \):
  Use \( g(-1) = (-1)^2 + 3(-1) - 10 = 1 - 3 - 10 = -12 \)

These points, when plotted along with the vertex and axis of symmetry, will help you accurately draw the parabola.
The more points you plot, the more accurate your graph will be. As you plot these points, make sure to check your symmetry; points on the left of the axis should have corresponding points on the right at the same \( y \)-value.