91Ó°ÊÓ

The vertex of a parabola is a very important point. It is the tip or the turning point of the curve in the parabola. In a quadratic function of the form \(ax^2 + bx + c\), the vertex can tell us a lot about the function. The x-coordinate of the vertex is calculated using the formula \( x = -\frac{b}{2a} \). Once you find the x-coordinate, plug it back into the function to find the corresponding y-coordinate of the vertex.
For example, in the function \( g(x) = 2x^2 - 8x + 7 \), you find the x-coordinate of the vertex by substituting \( a = 2 \) and \( b = -8 \) into the formula:

• \( x = -\frac{-8}{2 \times 2} = 2 \)

This means the vertex is located at \( x = 2 \). Plugging this into the function gives \( g(2) = -1 \), so the vertex of this parabola is at the point \((2, -1)\). This point provides crucial information because the y-coordinate represents either the maximum or minimum value of the parabola, which we'll explore next.

Every parabola has either a highest point or a lowest point, and this is its maximum or minimum value. In the case of a quadratic function like \( ax^2 + bx + c \), if the coefficient \( a \) is positive, the parabola opens upwards and has a minimum value at the vertex. If \( a \) is negative, it opens downwards and has a maximum value at the vertex.
In our function \( g(x) = 2x^2 - 8x + 7 \), the coefficient \( a = 2 \) is positive. This tells us that the parabola opens upwards and thus has a minimum value at the vertex. We've calculated that the vertex is at the point \((2, -1)\). This means the minimum value of the function is \(-1\), occurring when \( x = 2 \).
Understanding whether a quadratic function reaches a maximum or minimum at its vertex helps in graphing the function and analyzing real-world situations where such mathematical models apply.

A parabola is more than just a curve; it's the graphical representation of a quadratic equation. The shape of a parabola is determined by the quadratic function \( ax^2 + bx + c \), and key characteristics about it can be derived from the coefficients \( a \), \( b \), and \( c \).
This curve can be either facing upwards or downwards:

• If \( a \) is positive, the parabola opens upwards and has a U-like shape.
• If \( a \) is negative, the parabola opens downwards, forming an upside-down U.

The vertex is the point at which the parabola changes direction, forming the tip of the U or the highest/lowest point, depending on the orientation. The axis of symmetry, a vertical line that runs through the vertex, helps divide the parabola into two mirror-image halves.
In the provided function \( g(x) = 2x^2 - 8x + 7 \), the coefficients indicate that our parabola opens upwards and the vertex is at \((2, -1)\). Understanding these features provides a clear picture of how the function behaves and can guide solving various practical problems related to motion, profit maximization, or physics.