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When you look at the graph of a quadratic function, one of the first things you'll notice is its curvature, which is also referred to as its concavity. The function being considered here, \( f(x)=\frac{3}{2} x^2+6 x+4 \) shows what is called a 'parabolic' curve.

The curvature is determined by the sign of the leading coefficient, that is \( a \) in the general form \( ax^2 + bx + c \). If \( a > 0 \), the graph opens upwards, creating a 'concave-up' parabola. Conversely, if \( a < 0 \), the graph opens downwards, meaning it is 'concave-down'.

In our case, the coefficient \( a=\frac{3}{2} \) is positive, therefore the graph of \( f(x) \) is concave-up, indicating that the lowest point on the graph is the vertex. Understanding this concept assists in visualizing graph behavior, which in turn facilitates further analysis such as finding the maximum or minimum values the function can reach.

The vertex of a parabola is the 'tip' of the curve where the graph changes direction. In a concave-up parabola, the vertex represents the lowest point, and in a concave-down parabola, it is the highest point. The vertex is a useful point on the graph because it gives the maximum or minimum value of the quadratic function, which can be crucial in various applications such as optimization problems.

The vertex \( (h, k) \) of the parabola given by the general formula \( f(x) = ax^2 + bx + c \) can be found using the formulas \( h = -\frac{b}{2a} \) and \( k = f(h) \). This coordinates \( h \) and \( k \) represent, respectively, the x- and y-values of the vertex. For the function \( f(x)=\frac{3}{2}x^2+6x+4 \), the location of the vertex can be calculated using these formulas, and it is crucial for graphing the function accurately.

In the quadratic function \( f(x) = ax^2 + bx + c \), the values of \( a \) , \( b \) , and \( c \) are called coefficients. These coefficients are essential as they not only influence the curvature and the vertex, but also the width of the parabola and its position on the graph.

• The coefficient \( a \) determines the vertical stretch or compression of the graph. A larger absolute value of \( a \) makes the parabola narrower, while a smaller value makes it wider.
• The coefficient \( b \) affects the axis of symmetry of the parabola, which is a line that divides the graph into two mirror images. More specifically, it helps in finding the x-coordinate of the vertex.
• The coefficient \( c \) represents the y-intercept of the graph, which is where the graph crosses the y-axis.

Understanding these coefficients allows for a more detailed analysis of the quadratic function's graph, enabling you to sketch it accurately and to glean layers of information about the function's properties and how it behaves with different inputs.