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The vertex of a parabola is a very important point as it represents the parabola's highest or lowest point, depending on the direction it opens. For a quadratic function in the form \[ f(x) = ax^2 + bx + c \]the vertex \((h, k)\) can be found using the formula:- \( h = \frac{-b}{2a} \) to find the x-coordinate.- Once \( h \) is known, substitute \( x = h \) into the original function to find \( k \), which is the y-coordinate of the vertex.

For our example, the coefficients are \( a = 3 \), \( b = -6 \), and \( c = 7 \). Substituting these values gives:- \( h = \frac{-(-6)}{2 \times 3} = 1 \)- \( k = 3(1)^2 - 6(1) + 7 = 4 \)Thus, the vertex of the parabola is at \((1, 4)\). The parabola will have a minimum point here because it opens upwards, as explained by the positive \( a \) value.

The y-intercept is the point where the graphed line crosses the y-axis. For any quadratic function \[ f(x) = ax^2 + bx + c \]the y-intercept can be found where \( x = 0 \). This means substituting 0 for x in the function:- For our function: \( f(0) = 3(0)^2 - 6(0) + 7 = 7 \)

This calculation shows that the y-intercept of the parabola for this function is at the coordinate \((0, 7)\). The y-intercept is a crucial point because it provides a landmark for sketching the graph of the quadratic function. Since this point is above the x-axis and the parabola opens upward, it aids in understanding the parabola's direction.

X-intercepts are the points where the graph of the quadratic function crosses the x-axis. These points can be found by setting the function equal to zero: \[ ax^2 + bx + c = 0 \]and solving for \( x \). The quadratic formula:- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)is a reliable method for finding these intercepts.

In the given function, substituting the values \( a = 3 \), \( b = -6 \), and \( c = 7 \) into the formula shows that:- The discriminant \( b^2 - 4ac \) is calculated as \( 36 - 84 = -48 \).
Since the discriminant is negative, this quadratic function does not have real x-intercepts. This means the parabola does not cross the x-axis and is entirely above it, which aligns with the vertex and y-intercept that were found earlier.

The quadratic formula is crucial for solving quadratic equations. It turns complex problems into manageable solutions by providing a universal formula:- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \[ ax^2 + bx + c = 0 \].

This formula can yield:

• Two real solutions when the discriminant \( (b^2 - 4ac) \) is positive.
• One real solution when the discriminant is zero, indicating the parabola touches the x-axis at one point.
• No real solutions if the discriminant is negative, meaning the parabola does not touch the x-axis.

In our example, since the discriminant was \(-48\), there are no real x-intercepts for the parabola. This confirms what is observed from both the direction of the parabola and its vertex positioning.