91Ó°ÊÓ

The vertex of a parabola is a crucial point that represents the peak or the lowest point of the graph, depending on its orientation. For quadratic functions in the form \( f(x) = ax^2 + bx + c \), identifying the vertex helps in understanding the graph's shape and position. This is done using the vertex formula.
The vertex is given by the coordinates \((h, k)\), where:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

In the quadratic function \( f(x) = x^2 + 2x - 3 \), substitute \( a = 1 \) and \( b = 2 \) into the formula to find \( h = -\frac{2}{2} = -1 \). Then, calculate \( k \) by evaluating \( f(-1) \), giving \( k = -4 \). Therefore, the vertex is at \((-1, -4)\). This point tells us the parabola reaches its lowest point here, giving the graph a minimum.

X-intercepts are the points where the graph crosses the x-axis. These occur where \( f(x) = 0 \). Finding them involves solving the quadratic equation, which can often be done using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
For the equation \( x^2 + 2x - 3 = 0 \), use the values \( a = 1 \), \( b = 2 \), and \( c = -3 \) in the formula:

• Calculate the discriminant: \( b^2 - 4ac = 4 + 12 = 16 \)
• Solve for \( x \): \( x = \frac{-2 \pm 4}{2} \)

This computation results in \( x = 1 \) and \( x = -3 \), meaning the parabola meets the x-axis at \((1, 0)\) and \((-3, 0)\). These intercepts give important points where the graph truly intersects the axis and help trace the parabola's path.

The y-intercept of a function is where the graph crosses the y-axis. It's the point where \( x = 0 \). This is easier to compute compared to x-intercepts because you simply substitute \( x = 0 \) into the function.
For the quadratic function \( f(x) = x^2 + 2x - 3 \), evaluate \( f(0) \):

• Substitute: \( 0^2 + 2 \cdot 0 - 3 = -3 \)

Hence, the y-intercept is \((0, -3)\). This place on the graph shows where the curve passes through the y-axis. It serves as a reference point when sketching the graph, as it's a fixed point easy to find by inspection.

Graphing a quadratic equation like \( f(x) = x^2 + 2x - 3 \) involves plotting key points and connecting them smoothly. The critical points include the vertex, x-intercepts, and y-intercept. Here’s a step-by-step approach:
1. **Plot the Vertex**: Begin by plotting the vertex \((-1, -4)\) on the graph. This point is the turning or lowest point, where the direction of the parabola changes.

• The parabola opens upward because \( a = 1 \) is positive.

2. **Plot X-intercepts**: Mark \((1, 0)\) and \((-3, 0)\) on the x-axis. These points are where the graph intersects the x-axis.
3. **Plot the Y-intercept**: Add \((0, -3)\) to the plot. This is where the graph crosses the y-axis and gives further direction.
4. **Draw the Parabola**: Connect these points smoothly to form a U-shaped curve that opens upward. Ensure that the curve passes through all plotted points, giving a visual representation of the quadratic function.
By following these steps, you achieve a complete and accurate graph of the quadratic function, helping visual learners see the spatial relationships.