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Completing the square is a method used to transform a quadratic expression into a perfect square trinomial plus a constant. This technique helps in finding the vertex form of a quadratic equation, which makes it easier to identify the vertex of the parabola. In the quadratic function, such as \( f(x) = x^2 + 8x + 7 \), it can initially be hard to see the vertex just by inspection.
To complete the square, you must first ensure the coefficient of \( x^2 \) is 1. Fortunately, in our example, it already is. The next steps are:

• Take half of the coefficient of \( x \), square it, and add and subtract this square inside the quadratic expression.
• For \( x^2 + 8x \), half of 8 is 4, and squaring it gives 16, so you add and subtract 16: \( f(x) = (x^2 + 8x + 16) - 16 + 7 \).
• Rewrite it as a perfect square and simplify: \( f(x) = (x + 4)^2 - 9 \).

Now that it's in vertex form \( (x + 4)^2 - 9 \), the vertex \( (h, k) \) can be easily identified where \( h = -4 \) and \( k = -9 \). The vertex is \((-4, -9)\).
This form shows the vertex and makes sketching the parabola's graph straightforward.

The vertex formula is a quick way to find the vertex of a parabola described by the quadratic equation \( ax^2 + bx + c \). This method is often preferred because it's straightforward and doesn’t require rewriting the expression. For a function like \( f(x) = x^2 + 8x + 7 \), which is already in standard form, using the vertex formula can save you time.
The formula to find the x-coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 8 \), so:

• \( x = -\frac{8}{2 \times 1} = -4 \)

This tells us the x-coordinate of the vertex is \( -4 \). To find the y-coordinate, substitute \( x = -4 \) into the function:

• Calculate: \( f(-4) = (-4)^2 + 8(-4) + 7 = -9 \)

Therefore, the y-coordinate is \( -9 \). Thus, the vertex is located at \((-4, -9)\).
The vertex formula swiftly leads you to the vertex without additional computations, making it efficient for parabolas in standard form.

A quadratic function graph is a type of U-shaped curve known as a parabola. The quadratic equation \( f(x) = x^2 + 8x + 7 \) portrays this with its graph showing symmetry around its vertex. By identifying the vertex and the axis of symmetry, you can easily sketch the graph.
The vertex of a quadratic function \( (h, k) \) is not just a point on the graph. It tells us where the parabola changes direction. In our specific example, \((-4, -9)\) is the lowest point (unless the parabola is upside down, which applies to functions with a negative leading coefficient).
Here are some crucial characteristics of a quadratic graph:

• The axis of symmetry is a vertical line through the vertex, for \( x = -4 \) in our example.
• The direction of the parabola's opening is upward because the \( x^2 \) term is positive.
• The vertex is the minimum point of the graph for upward-facing parabolas.

Knowing these features allows you to draw the parabola accurately by plotting the vertex, direction of opening, and symmetry, giving a clear visual representation of the function.