Question

True or False The graph of \(f(x)=2 x^{2}+3 x-4\) is concave up.

Short answer

True

Step-by-step solution

Understand the function

The given function is a quadratic function: \(f(x) = 2x^2 + 3x - 4\). In general, a quadratic function is of the form \(ax^2 + bx + c\).

Identify the coefficient of the quadratic term

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

Determine concavity

For a quadratic function \(ax^2 + bx + c\), the graph is concave up if the coefficient \(a > 0\). If \(a < 0\), the graph is concave down. Since \(a = 2\), which is greater than zero, the graph is concave up.

Key concepts

Concavity

In mathematics, concavity refers to the direction in which a curve opens. For a quadratic function like \(f(x) = 2x^2 + 3x - 4\), the concavity is determined by the sign of the coefficient of the quadratic term. If the coefficient \(a\) is positive, the graph opens upwards, resembling a U-shape. This means the function is concave up. On the other hand, if \(a\) is negative, the graph opens downwards, taking an inverted U-shape, and the function is concave down. Since in our exercise \(a = 2\), the function is concave up because 2 is greater than zero.

Coefficient

A coefficient in a quadratic function is the number in front of a variable term and it dictates the shape and position of the graph. In the quadratic function \(f(x) = 2x^2 + 3x - 4\), there are three coefficients: 2 is the coefficient of \(x^2\) term, 3 is the coefficient of \(x\) term, and -4 is the constant term. The most critical coefficient for determining concavity is the one in front of the \(x^2\) term, which is 2 in this case. It tells us that the graph will be concave up because it is positive. Understanding these coefficients can help predict the behavior and appearance of the graph.

Parabola

A parabola is the graph of a quadratic function. It is a symmetric curve that either opens upward or downward. For the function \(f(x) = 2x^2 + 3x - 4\), the graph forms a parabola that opens upward due to the positive coefficient of the \(x^2\) term. Parabolas have several important features, such as the vertex (the highest or lowest point), the axis of symmetry (a vertical line that divides the parabola into two mirror images), and the direction of opening (concave up or concave down). Knowing how to identify these features allows students to better understand and graph quadratic functions.

Graphing

Graphing a quadratic function involves several steps which help visualize the function's behavior. Start by identifying the coefficients and determining the direction of the parabola's opening (concavity). For \(f(x) = 2x^2 + 3x - 4\), we know it opens upward since the coefficient of \(x^2\) is positive. Next, find the vertex using the formula \(x = -\frac{b}{2a}\). For our function, the vertex is at \(x = -\frac{3}{2(2)} = -\frac{3}{4}\). Then, calculate the y-value by substituting this x-value back into the original equation. Plot the vertex, find additional points by choosing x-values, and finally draw the curve connecting these points. This process helps students comprehend the shape and position of the quadratic graph thoroughly.