91Ó°ÊÓ

Quadratic functions often form a u-shaped graph called a parabola. The vertex is a crucial part of this graph, representing either the lowest or highest point, depending on whether the parabola opens upward or downward.
The vertex of a parabola can be found using the vertex formula, which is helpful for graphing and analyzing quadratic functions. Given a quadratic equation in standard form, \( f(x) = ax^2 + bx + c \), the vertex formula allows us to find the x-coordinate of the vertex using:

• \( x = -\frac{b}{2a} \)

After determining the x-coordinate, we can find the y-coordinate by plugging it back into the original equation.
In the given function, \( f(x) = 2x^2 - 8x + 6 \), we calculate \( x = -\frac{-8}{2 \times 2} = 2 \) and use it to find \( y \):

• \( f(2) = 2(2)^2 - 8(2) + 6 = -2 \)

This gives us the vertex of (2, -2). Knowing the vertex helps in predicting the direction and the shape of the parabola.

Intercepts are points where the graph crosses the axes, and they help in sketching the parabola accurately.
The y-intercept is the point where the graph crosses the y-axis. To find it, simply substitute \( x = 0 \) in the equation:

• \( f(0) = 6 \)

Here, the y-intercept is at point (0, 6).
X-intercepts are more involved, representing where the graph crosses the x-axis. They are found by setting the equation to zero and solving for \( x \):

• \( 2x^2 - 8x + 6 = 0 \)

Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we solve:

• \( x = \frac{8 \pm \sqrt{16}}{4} \)

Giving us the x-intercepts at (1, 0) and (3, 0).
Intercepts provide a clear picture of how the curve interacts with the coordinate system, making them essential for graphing.

Graphing a parabola involves plotting critical points on a coordinate plane and drawing a curve through them. Start with the vertex, intercepts, and additional points to map out the shape.
For function \( f(x) = 2x^2 - 8x + 6 \), plot the vertex (2, -2), y-intercept (0, 6), and x-intercepts (1, 0) and (3, 0). Choosing extra points like (4, 8) ensures a more detailed graph.
The parabola opens upward since \( a = 2 > 0 \), forming a U shape. Here's a step-by-step to graph:

• Plot point (2, -2), the vertex, marking the center of the parabola.
• Add the intercepts: (0, 6), (1, 0), and (3, 0).
• Plot additional point (4, 8) to draw the curve smoothly.

Connect these points with a smooth, continuous curve. Ensure that the curve reflects the symmetry around the vertex.