Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The functions \(f(x)=3 x^{2}+6 x+7 \quad\) and \(g(x)=3 x^{2}+6 x-1\) have the same vertex.

Step-by-Step Solution

Answer

The statement that the functions \(f(x)=3 x^{2}+6 x+7 \) and \(g(x)=3 x^{2}+6 x-1\) have the same vertex is False.

Step 1: Identify Coefficients

For \(f(x)\), \(a = 3\), \(b = 6\), \(c = 7\); for \(g(x)\), \(a = 3\), \(b = 6\), \(c = -1\)

Step 2: Calculate Vertices

Calculate the x-coordinate of the vertex of \(f(x)\) by plugging \(a\) and \(b\) into the formula: \(-b/2a\). For \(f(x)\), it is \(-6/(2*3) = -1\). Do the same for \(g(x)\) which also yields -1. The functions \(f(x)\) and \(g(x)\) both have a vertex with the same x-coordinate.

Step 3: Conclusion

Having confirmed that \(f(x)\) and \(g(x)\) have identical x-coordinates for their vertices, we still need to inform whether the y-coordinates (resulting from substituting x-coordinate of the vertex back into the functions) are identical to conclusively verify that the vertex of the two functions are identical. Substituting \(x = -1\) into \(f(x)\) yields \(3 * (-1)^2 + 6*-1 + 7 = 4\) and into \(g(x)\) yields \(3 * (-1)^2 + 6*-1 - 1 = -4\). The y-coordinates of the vertices are not identical, hence the vertex of the two functions is not identical and the statement is False.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental to algebra and represent the simplest form of polynomials with a degree of two. They are commonly written in the standard form, just like you've seen with the functions f(x) = 3x^2 + 6x + 7 and g(x) = 3x^2 + 6x - 1 given in your exercise.

The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants, and the highest exponent of x is 2. The coefficient a is particularly important as it determines the parabola's direction of opening — if a is positive, the parabola opens upwards and if negative, it opens downwards.

Understanding the structure of quadratic functions allows us to predict their graph's shape and characteristics, such as the vertex, axis of symmetry, and the y-intercept. For instance, both functions in the exercise have the same a and b values, which will influence their vertices and symmetry.

Vertex Formula

The vertex formula is an invaluable tool for finding the turning point of a parabola, which is the highest or lowest point on the graph depending on whether it opens up or down. It's derived from the quadratic function's standard form and guides us to the parabola's vertex.

The x-coordinate of a parabola's vertex can be found with the formula x = -b/(2a). This equation gives us the axis of symmetry for the parabola. In our exercise, for both functions f(x) and g(x), you have calculated this x-value to be -1 using this formula. Despite the differing c values, the x-coordinate of the vertex remained unaffected.

Next, to find the y-coordinate of the vertex, you substitute the x-value back into the original quadratic equation. This final coordinate determines the vertex's actual position on the graph. In this instance, despite sharing the same x-coordinate for the vertex, the functions do not share the same y-coordinate, resulting in them having different vertices.

Parabolas

Parabolas are the geometric representations of quadratic functions and are U-shaped curves. The most notable point of a parabola is its vertex, which is the peak or trough of the curve and acts as a mirror line (axis of symmetry) for the shape. When you graph a quadratic function, the vertex's coordinates will tell you where this unique point of the parabola lies on the Cartesian plane.

For any quadratic function, the shape of the parabola depends on the coefficient a: if a is positive, the parabola opens upward, if it's negative, downward. However, it's the combination of the coefficients a, b, and c that determine the exact location and size of the parabola. Your exercise highlights a key aspect of parabolas: two functions can have a parabola with the same curvature and axis of symmetry but have different vertices due to differing values in their constant term, c. This demonstrates that even small changes in the quadratic equation can lead to significant differences in the graph's appearance.