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To find the vertex of a quadratic function, a handy tool is the vertex formula. This formula helps pinpoint the highest or lowest point on the graph, known as the vertex. For a quadratic function written in the form \( ax^2 + bx + c \), the vertex \((h, k)\) can be found using:

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

The first step, finding \( h \), calculates the \( x \)-coordinate of the vertex. Simply substitute the values of \( a \) and \( b \) from the function into the formula. In our example, with \( a = 5 \) and \( b = -10 \), this gives:\[h = -\frac{-10}{2 \times 5} = 1\]Next, to find \( k \), substitute \( h \) back into the original quadratic function. In this case:\[ k = f(1) = 5(1)^2 - 10(1) + 3 = -2\]Thus, the vertex of the quadratic function \( f(x) = 5x^2 - 10x + 3 \) is \((1, -2)\). Identifying the vertex is crucial for understanding the behavior of the quadratic graph and is also used in other mathematical applications.

Identifying coefficients is a straightforward but essential part of working with quadratic functions. The standard form of a quadratic function is \( ax^2 + bx + c \). Here:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In the function \( f(x) = 5x^2 - 10x + 3 \), these values are:

• \( a = 5 \)
• \( b = -10 \)
• \( c = 3 \)

These coefficients are crucial for many operations:- They determine the direction the parabola opens (upward if \( a > 0 \), downward if \( a < 0 \)).- They play a role in calculating the vertex and the axis of symmetry.Understanding \( a \), \( b \), and \( c \) helps to quickly analyze any quadratic function and anticipate the graph's general shape, direction, and vertex location.

Graphing a quadratic function brings all the elements together to visualize the equation. The graph of a quadratic function is a parabola, which can open upwards or downwards, depending on the sign of the coefficient \( a \). Here’s how graphing fits with vertex and coefficients.

• The vertex \((h, k)\) identifies the parabola's turning point or peak.
• The coefficient \( a \) indicates the parabola's direction and "width." A larger \( |a| \) makes it narrower, while a smaller \( |a| \) makes it wider.
• The line of symmetry is vertical, passing through \( x = h \).

Using our example \( f(x) = 5x^2 - 10x + 3 \), the identified vertex \((1, -2)\) shows where the graph turns. Since \( a = 5 \), the parabola opens upwards and is relatively narrow. Knowing how to identify these parts means you can sketch the graph of any quadratic function by hand, quickly marking points like the vertex and reflecting about its axis.