Problem 37 - College Algebra

Question

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=-4 x^{2}+8 x-3\)

Step-by-Step Solution

Answer

The given quadratic function has a maximum value. The coordinates of the maximum point are (1, 1).

Step 1: Identify the coefficients

In the quadratic function \( f(x) = -4x^2 + 8x -3 \), -4 is the coefficient \( a \), 8 is the coefficient \( b \), and -3 is the coefficient \( c \).

Step 2: Determine the maximum or minimum value

The given function has a coefficient \( a \) of -4, which is less than zero. Therefore, the quadratic function has a maximum value.

Step 3: Find the x-coordinate of the maximum point

To find the x-coordinate of the maximum point, we use the formula \( x = -b/2a \). Substituting -4 for \( a \) and 8 for \( b \), we find that \( x = -8 / (2*-4) = 1 \).

Step 4: Find the y-coordinate of the maximum point

Substitute \( x = 1 \) into the equation \( f(x) = -4x^2 + 8x -3 \) to find the y-coordinate of the maximum point. The y-coordinate is given by \( f(1) = -4(1)^2 + 8(1) -3 = 1 \).

Key Concepts

Coefficients in Quadratics

In the world of quadratic functions, understanding coefficients is crucial. A quadratic function is typically written in the form \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are known as coefficients. Each one has a distinct role:

• \( a \): Determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. If negative, it opens downwards.
• \( b \): Influences the position of the axis of symmetry and hence, the location of the vertex.
• \( c \): Indicates the y-intercept, which is where the graph crosses the y-axis.

In the function given in our exercise, \( f(x) = -4x^2 + 8x -3 \), the coefficients are \( a = -4 \), \( b = 8 \), and \( c = -3 \). Knowing that \( a = -4 \), we can quickly deduce that the parabola opens downwards, signaling the presence of a maximum point instead of a minimum.

Maximum and Minimum Points

Identifying whether a quadratic function has a maximum or a minimum point hinges largely on the coefficient \( a \). For any quadratic function represented as \( ax^2 + bx + c \):

• If \( a > 0 \), the parabola opens upwards, and the function has a minimum point.
• If \( a < 0 \), the parabola opens downwards, and it has a maximum point.

In our example, since \( a = -4 \) (which is less than zero), the function \( f(x) = -4x^2 + 8x -3 \) exhibits a maximum point. This is because the negative value of \( a \) causes the arms of the parabola to point downwards, creating a peak or a maximum value.

Vertex Formula

The vertex of a quadratic function, which is either the maximum or minimum point, can be found using the vertex formula. The x-coordinate of the vertex is given by the formula \( x = -\frac{b}{2a} \). This formula is derived from completing the square technique or from calculus, and helps locate the axis of symmetry.
For our function, \( f(x) = -4x^2 + 8x -3 \), substituting \( a = -4 \) and \( b = 8 \) into the vertex formula gives:\[ x = -\frac{8}{2 \times -4} = 1 \]Once the x-coordinate is determined, the y-coordinate can be found by substituting \( x \) back into the original function. Hence:\[ f(1) = -4(1)^2 + 8(1) -3 = 1 \]So, the vertex of the function is at the point \((1, 1)\), identifying it as the function's maximum point.