Problem 17 - Algebra for College Students

Question

Determine whether the graph of each quadratic function opens upward or downward. $$f(x)=(-2 x+3)^{2}$$

Step-by-Step Solution

Answer

The parabola opens upward.

Step 1: Simplify the quadratic function

First, expand the quadratic function if necessary. The given function is \[f(x) = (-2x + 3)^2\]. We need to expand this expression to identify the coefficient of the \(x^2\) term.

Step 2: Expand the square

To expand \((-2x + 3)^2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = -2x\) and \(b = 3\). So, \[ (-2x + 3)^2 = (-2x)^2 + 2(-2x)(3) + 3^2\]. Simplify to get:\[ 4x^2 - 12x + 9\].

Step 3: Identify the quadratic term’s coefficient

The expanded function is \[ f(x) = 4x^2 - 12x + 9\]. The coefficient of the \(x^2\) term is \(4\).

Step 4: Determine the direction of the parabola

If the coefficient of the \(x^2\) term is positive, the parabola opens upward. If it is negative, the parabola opens downward. In this case, the coefficient is \(4\).

Step 5: Conclusion

Since the coefficient \(4\) is positive, the graph of the function \(f(x) = (-2x + 3)^2\) opens upward.

Key Concepts

Parabolas

A parabola is a U-shaped curve that is the graph of a quadratic function. Quadratic functions take the general form \[ f(x) = ax^2 + bx + c \], where \(a\), \(b\), and \(c\) are constants. The value of \(a\) determines the width and direction of the parabola, while \(b\) and \(c\) affect its position on the graph. Parabolas have a single vertex, which is either the highest or lowest point on the graph, depending on the direction they open. There is also an axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

Coefficient of x^2 Term

In the quadratic function, the term containing \(x^2\) is crucial because its coefficient, \(a\), directly affects the shape and direction of the parabola. The general form of a quadratic function is \[ f(x) = ax^2 + bx + c \].
The coefficient \(a\) can tell us several things:

• If \(a > 1\), the parabola becomes narrower compared to the standard parabola \( y = x^2 \).
• If \(0 < a < 1\), the parabola becomes wider.
• If \(a < 0\), the parabola flips and opens downward.
• If \(a = 0\), the function is not quadratic but linear.

Direction of Parabola

The direction in which a parabola opens is determined by the sign of the coefficient of the \(x^2\) term. If the coefficient is positive, the parabola opens upward. If it is negative, the parabola opens downward. To determine the direction, follow these steps:

• Identify the coefficient of the \(x^2\) term in the quadratic function.
• If the coefficient is positive \((a > 0)\), the parabola opens upward.
• If the coefficient is negative \((a < 0)\), the parabola opens downward.

For instance, in the function \[ f(x) = (-2x + 3)^2 \], after expanding, we get \[ 4x^2 - 12x + 9 \]. Here, the coefficient of the \(x^2\) term is 4, which is positive. Thus, the parabola opens upward.