Problem 15 - Algebra for College Students

Question

Determine whether the graph of each quadratic function opens upward or downward. $$y=-3 x^{2}+4 x+2$$

Step-by-Step Solution

Answer

The graph opens downward.

Step 1: Identify the quadratic function's general form

The general form of a quadratic function is given by: \[ y = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants.

Step 2: Determine the value of the coefficient \(a\)

From the given quadratic function: \[ y = -3x^2 + 4x + 2 \] Identify the coefficient \(a\). In this case, \(a = -3\).

Step 3: Analyze the coefficient \(a\)

Determine the direction in which the graph opens based on the value of \(a\): - If \(a > 0\), the graph opens upward. - If \(a < 0\), the graph opens downward. Since \(a = -3 < 0\), the graph opens downward.

Key Concepts

graph analysis

Graph analysis is essential when studying quadratic functions, as it reveals the shape and behavior of the graph. For any quadratic function, the graph is a parabola. To determine whether the parabola opens upward or downward, look at the coefficient of the \(x^2\) term, known as 'a'. \[ y = ax^2 + bx + c \] If \(a > 0\), the parabola is concave up (opens upward), resembling a U-shape. If \(a < 0\), the parabola is concave down (opens downward), resembling an upside-down U-shape. Remember, the sign and value of 'a' heavily influence the graph's orientation. When you analyze a given quadratic function, you can quickly predict the graph's shape by checking if 'a' is positive or negative.

coefficients

Coefficients play a crucial role in determining the characteristics of quadratic functions. In the general form of a quadratic equation: \[ y = ax^2 + bx + c \] the terms 'a', 'b', and 'c' are coefficients. Each coefficient affects the graph in different ways:

• The coefficient 'a' influences the direction of the parabola. If 'a' is positive, the parabola opens upward; if negative, it opens downward.
• The coefficient 'b' affects the slope and the x-intercepts of the quadratic function.
• The coefficient 'c' represents the y-intercept, indicating where the parabola crosses the y-axis.

Understanding these coefficients and how they contribute to the overall form and position of the graph is vital for graph analysis and solving quadratic equations.

general form of quadratic equation

The general form of a quadratic equation is \[ y = ax^2 + bx + c \] where:

• 'a' is the coefficient of the quadratics term \(x^2\)
• 'b' is the coefficient of the linear term \(x\)
• 'c' is the constant term

This form is fundamental for expressing quadratic functions. It's used for analyzing their graphs, solving quadratic equations, and understanding various properties. When given any quadratic function, rewrite it in this general form to easily identify each coefficient and predict the function’s behavior and graph characteristics. Always start by identifying 'a', 'b', and 'c' for a clear analysis.