Question

MAKING AN ARGUMENT The point (1, 5) lies on the graph of a quadratic function with axis of symmetry x = ?1. Your friend says the vertex could be the point (0, 5). Is your friend correct? Explain.

Short answer

No, your friend is not correct. The vertex of the function cannot be at the point (0,5) because the x-coordinate of the vertex should be the same as that of the axis of symmetry, which is -1, not 0.

Step-by-step solution

Understanding Quadratic Function

Quadratic functions have a general form \(f(x) = a(x-h)^2 + k\). Where (h,k) is the vertex of the function. The line of symmetry or axis of symmetry for the quadratic function is the vertical line x = h, where h is the x-coordinate of the vertex.

Apply our understanding to the problem

Given, the axis of symmetry of the function is x = -1. Therefore, the x-coordinate of the vertex is -1. But we are considering if the vertex could be the point (0,5). This is the point where the x-coordinate is 0.

Conclusion

Therefore, the point (0,5) cannot be the vertex of this function because the x-coordinate of the vertex is not equal to the x-coordinate of the axis of symmetry.

Key concepts

Axis of Symmetry

Imagine slicing a parabola precisely down the middle so that each side is a mirror image of the other. This imaginary line is known as the axis of symmetry. In quadratic functions of the form \( f(x) = a(x-h)^2 + k \), the axis of symmetry always passes through the vertex of the parabola and is represented by the equation \( x = h \). Where \( h \) signifies the x-coordinate of the vertex.

For instance, if we know that a quadratic function has its axis of symmetry at \( x = -1 \), we can immediately ascertain that the x-coordinate of the vertex must also be -1. This line divides the parabola into two symmetrical halves, reflecting the pattern of the quadratic function's graph. Understanding this can be crucial when attempting to graph the function or locate the vertex, as it provides a clear direction where to focus.

Vertex of a Quadratic

The vertex of a quadratic function is the single point that represents the peak of its graph, whether it points upwards (minimum value) or downwards (maximum value). In the standard quadratic function equation \( f(x) = a(x-h)^2 + k \), the vertex can be found at the point (h, k).

Let's apply this to a real-life scenario. If your friend suggests that the vertex could be at the point (0, 5), recall that the vertex's x-coordinate must align with the axis of symmetry. In our example, since the axis of symmetry is at \( x = -1 \), the x-coordinate of the vertex also needs to be -1. Consequently, the vertex cannot be at (0, 5) as posited by your friend. Instead, it must be somewhere along the line \( x = -1 \), which aids significantly when sketching the graph or finding other characteristics of the quadratic function.

Quadratic Equation

A quadratic equation typically appears in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. When you graph a quadratic equation, you get a parabola. Sometimes, you may need to solve for the variable, which can be achieved by factoring, completing the square, using the quadratic formula, or graphically.

Effective understanding of quadratic equations goes hand-in-hand with recognizing their implications on graphs. For example, knowing the equation can help decipher the location of the function's vertex and whether it opens up or down, which hinges on the sign of \( a \). Solving these equations can lead you to find the function's roots or zeros, which are the x-intercepts of the parabola, adding yet another layer of significance to mastering quadratic equations for both algebraic and graphical analysis.