Problem 16 - College Algebra

Question

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=-x^{2}+4 x+3 $$

Step-by-Step Solution

Answer

The function has a maximum value of 7 at \( x = 2 \), with the axis of symmetry also at \( x = 2 \).

Step 1: Identify the form of the quadratic

The function given is \( f(x) = -x^2 + 4x + 3 \). This is in the standard form of a quadratic equation, \( f(x) = ax^2 + bx + c \), where \( a = -1 \), \( b = 4 \), and \( c = 3 \).

Step 2: Find the direction of the parabola

Since \( a = -1 \) is negative, the parabola opens downward, indicating that there is a maximum value for this quadratic function.

Step 3: Determine the axis of symmetry

The axis of symmetry for a quadratic function \( f(x) = ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting the values of \( b \) and \( a \) into this formula, we have \( x = -\frac{4}{2(-1)} = 2 \). So, the axis of symmetry is \( x = 2 \).

Step 4: Find the maximum value

To find the maximum value, substitute \( x = 2 \) back into the function \( f(x) = -x^2 + 4x + 3 \). Calculating this gives: \[ f(2) = -(2)^2 + 4(2) + 3 = -4 + 8 + 3 = 7. \] Thus, the maximum value is 7.

Key Concepts

Maximum Value of a Quadratic Function

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.

Axis of Symmetry

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 \( x = 2 \).

Standard Form of Quadratic Equation

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.