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Quadratic functions can either have a minimum or maximum value based on the coefficient of the squared term, denoted as \( a \). If \( a \) is positive, the parabola opens upwards and has a minimum value. Conversely, if \( a \) is negative, the parabola opens downwards and has a maximum value. In our function, \( f(x) = -x^2 + 4x + 3 \), \( a = -1 \), which is negative. This tells us the parabola opens downwards, hence there is a maximum value.

To find this maximum value, you substitute the x-coordinate of the axis of symmetry back into the function. Here, we calculated that at \( x = 2 \), the function gives us \( f(2) = 7 \). Thus, this is the highest point on the graph of the quadratic function, making 7 the maximum value.

The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two mirror-image halves. It's a crucial feature for determining the vertex of the parabola, where either the minimum or maximum value occurs.

For any quadratic function written in the form \( f(x) = ax^2 + bx + c \), the formula to find the axis of symmetry is \( x = -\frac{b}{2a} \).

• Substitute \( b = 4 \) and \( a = -1 \) into this formula: \( x = -\frac{4}{2(-1)} \).
• Simplifying gives \( x = 2 \). So the line \( x = 2 \) is the axis of symmetry for this function.

This means the parabola is symmetrically balanced around the vertical line \( x = 2 \).

A quadratic equation can be expressed in the standard form: \( f(x) = ax^2 + bx + c \). This format is particularly helpful as each coefficient gives us essential information about the parabola it graphs.

• The coefficient \( a \) determines the direction of the parabola (upwards if \( a > 0 \), downwards if \( a < 0 \)).
• The coefficient \( b \) helps in finding the axis of symmetry.
• The constant \( c \) gives the y-intercept, where the graph crosses the y-axis.

Understanding the standard form allows you to quickly identify these elements and solve related problems effectively. In our case, \( f(x) = -x^2 + 4x + 3 \) is already in standard form, facilitating the determination of the parabola's characteristics such as the axis of symmetry and maximum value.