Understanding Quadratic Functions Vertex At Y-intercept

Let's delve into the characteristics of quadratic functions, particularly those where the vertex coincides with the y-intercept. Understanding the interplay between a quadratic function's vertex, y-intercept, and axis of symmetry is crucial for solving this problem. A quadratic function, generally expressed in the form f(x) = ax² + bx + c, exhibits a parabolic shape. The vertex of this parabola represents either the minimum or maximum point of the function. The y-intercept is the point where the parabola intersects the y-axis, which occurs when x = 0. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Let's analyze the given scenario where the vertex and y-intercept are the same point. This implies that the parabola's lowest or highest point lies on the y-axis. Consider the standard form of a quadratic equation, f(x) = ax² + bx + c. The y-intercept is found by setting x = 0, which gives us f(0) = c. Thus, the y-intercept is the point (0, c). The vertex of a parabola in the form f(x) = ax² + bx + c is given by the coordinates (-b/2a, f(-b/2a)). If the vertex is the same as the y-intercept, then the x-coordinate of the vertex must be 0. This means -b/2a = 0, which implies that b = 0. Therefore, our quadratic function simplifies to f(x) = ax² + c. This form tells us a great deal about the parabola's symmetry. Since there is no bx term, the parabola is symmetric about the y-axis. The axis of symmetry is a vertical line that passes through the vertex. Since the vertex lies on the y-axis, the axis of symmetry is the line x = 0. This is a key piece of information.

The question asks which statement must be true. Option A states, "The axis of symmetry for the function is x = 0." As we've deduced, when the vertex and y-intercept coincide, the axis of symmetry is indeed x = 0. This statement aligns perfectly with our analysis. Option B suggests, "The axis of symmetry for the function is y = 0." This is incorrect. The axis of symmetry is a vertical line, defined by an equation of the form x = k, where k is a constant. y = 0 represents the x-axis, which is a horizontal line. Therefore, Option B is false. Option C states, "The function has no x-intercepts." This isn't necessarily true. The existence of x-intercepts depends on the values of a and c in the equation f(x) = ax² + c. If a and c have opposite signs, the parabola will intersect the x-axis at two points. For instance, if f(x) = x² - 4, the vertex is at (0, -4) (which is the y-intercept), and the x-intercepts are at x = 2 and x = -2. However, if a and c have the same sign and c is not zero, or if c is positive and a is positive, the parabola will not intersect the x-axis. For example, if f(x) = x² + 4, there are no real x-intercepts. Therefore, the function may have x-intercepts, but it is not a certainty. Finally, let's consider Option D, which we'll assume is a placeholder for another possible statement. Based on our analysis, the most definitive and universally true statement is that the axis of symmetry is x = 0. This condition arises directly from the vertex and y-intercept being the same point, forcing the quadratic to be symmetrical about the y-axis. In conclusion, a quadratic function whose vertex is the same as its y-intercept must have an axis of symmetry at x = 0. This is a direct consequence of the function's symmetrical nature when its extreme point lies on the y-axis. Therefore, the correct answer is A.

Analyzing the implications of the Vertex Coinciding with the Y-intercept

When we consider a quadratic function where the vertex and the y-intercept are the same, we unlock a specific set of characteristics that the function must possess. This condition significantly narrows down the possibilities and allows us to make definitive statements about its properties. As previously established, a quadratic function can be generally written as f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The y-intercept is the point where the function crosses the y-axis, which occurs when x = 0. Substituting x = 0 into the equation, we find that the y-intercept is (0, c). The vertex of a parabola represented by f(x) = ax² + bx + c is located at the point (-b/2a, f(-b/2a)). If the vertex coincides with the y-intercept, then the x-coordinate of the vertex must be 0. This crucial condition leads us to the equation -b/2a = 0. Solving for b, we find that b = 0. This is a pivotal realization. When b = 0, our quadratic function simplifies to f(x) = ax² + c. This simplified form reveals the symmetry inherent in the function. The absence of the bx term signifies that the parabola is symmetric about the y-axis. The axis of symmetry is a vertical line that passes directly through the vertex. Since the vertex is on the y-axis, the axis of symmetry must be the line x = 0. This is a fundamental property of such quadratic functions. Understanding this, let's revisit the original question and the options provided. We are looking for a statement that must be true when the vertex and y-intercept are the same. We've established that the axis of symmetry is x = 0. Therefore, Option A, which states, "The axis of symmetry for the function is x = 0," is the correct answer. Now, let’s consider why the other options are not universally true. Option B, "The axis of symmetry for the function is y = 0," is incorrect. The axis of symmetry is a vertical line, represented by an equation of the form x = k, not a horizontal line like y = 0. Option C, "The function has no x-intercepts," is not necessarily true. While it is possible for the function to have no x-intercepts, it can also have one or two x-intercepts depending on the values of a and c. For example, if a is positive and c is negative, the parabola will open upwards and intersect the x-axis at two points. Conversely, if a and c are both positive, the parabola will not intersect the x-axis. Thus, the absence of x-intercepts is not a guaranteed characteristic.

Exploring x-intercepts and the Discriminant

The existence and nature of x-intercepts are closely related to the discriminant of the quadratic equation. The discriminant, denoted as Δ (delta), is a part of the quadratic formula and is given by the expression Δ = b² - 4ac. In our simplified quadratic function, f(x) = ax² + c, where b = 0, the discriminant becomes Δ = 0² - 4ac = -4ac. The discriminant provides valuable information about the roots (x-intercepts) of the quadratic equation: If Δ > 0, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points. If Δ = 0, the equation has one real root (a repeated root), meaning the vertex of the parabola touches the x-axis. If Δ < 0, the equation has no real roots, meaning the parabola does not intersect the x-axis. In our case, Δ = -4ac. The sign of the discriminant depends entirely on the signs of a and c. If a and c have opposite signs (one positive and one negative), then Δ = -4ac will be positive, indicating two distinct real roots (two x-intercepts). For example, consider the function f(x) = x² - 4. Here, a = 1 and c = -4. The discriminant is Δ = -4(1)(-4) = 16, which is positive. This function has two x-intercepts at x = 2 and x = -2. If a and c have the same sign (both positive or both negative), then Δ = -4ac will be negative, indicating no real roots (no x-intercepts). For example, consider the function f(x) = x² + 4. Here, a = 1 and c = 4. The discriminant is Δ = -4(1)(4) = -16, which is negative. This function has no real x-intercepts. If c = 0, then Δ = 0, indicating one real root. In this case, the function is f(x) = ax², and the vertex and the x-intercept are both at the origin (0, 0). This analysis confirms that the statement "The function has no x-intercepts" is not universally true for quadratic functions whose vertex is the same as its y-intercept. The presence of x-intercepts depends on the specific values of a and c. Therefore, we can confidently eliminate Option C as a possible answer. Our comprehensive examination leads us back to Option A, "The axis of symmetry for the function is x = 0," as the only statement that must be true. This characteristic is a direct and unavoidable consequence of the vertex coinciding with the y-intercept, making the parabola symmetric about the y-axis.

Conclusion: The Definitive Property of Symmetry

In conclusion, when a quadratic function has its vertex at the same point as its y-intercept, a crucial property emerges: the axis of symmetry must be the line x = 0. This arises from the inherent symmetry of parabolas and the specific constraints imposed by the vertex and y-intercept alignment. The y-intercept being at the same point as the vertex forces the x-coordinate of the vertex to be 0, which directly dictates that the axis of symmetry is the vertical line x = 0. While the existence of x-intercepts may vary depending on the coefficients of the quadratic function, the axis of symmetry at x = 0 remains a constant and definitive characteristic. This understanding solidifies Option A as the correct answer to the question, highlighting the importance of recognizing the fundamental relationships between a quadratic function's key features. Therefore, the correct answer is A. The axis of symmetry for the function is x=0. This is the only statement that must be true given the condition that the vertex and y-intercept are the same point. Other characteristics, such as the presence of x-intercepts, are contingent on the specific values of the coefficients in the quadratic equation.