Characteristics of Quadratic Functions Name Per Date Determine the value of the zeros, the equation of the axis of symmetry, the max or min value and the vertex 3 1. 2. Zaros: Zeros: Axis of symmetry Axis of symmetry: Zeros (4,0) 100 Axis of symmetry Xca Max or Min: Y:4 Vertex V: V:(-2/4) Max or
The Correct Answer and Explanation is :
To solve this quadratic function worksheet, let’s break it down step by step for each problem. Here’s how we can approach it:
Problem 1:
Given that the zeros of the quadratic function are (4,0)(4, 0) and (−1,0)(-1, 0), we can identify the key characteristics of the quadratic function.
- Zeros (Roots): The zeros of the function are the points where the graph intersects the x-axis. These points are the x-values for which the function equals zero. For this problem, the zeros are given as x=4x = 4 and x=−1x = -1.
- Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the midpoint between the two zeros.
- To find the axis of symmetry, take the average of the x-values of the zeros:
- Max or Min: Since the function opens upwards (the coefficient of x2x^2 is positive), the vertex represents the minimum value of the function. This minimum value occurs at the axis of symmetry.
- Vertex: The vertex is located at the point where the function reaches its maximum or minimum. The x-coordinate of the vertex is the same as the axis of symmetry, which is x=1.5x = 1.5.
- To find the y-coordinate of the vertex, we substitute x=1.5x = 1.5 into the quadratic equation. However, since we are given the zeros, we can also infer that the vertex is halfway between them, so we can approximate the vertex as (1.5,y)(1.5, y), where yy is calculated based on the quadratic function (which can be found if we expand the equation).
Problem 2:
For this case, the zeros are (4,0)(4, 0) and (−1,0)(-1, 0), and the axis of symmetry is x=1.5x = 1.5. Let’s assume that the quadratic function is in the form f(x)=a(x−4)(x+1)f(x) = a(x – 4)(x + 1). Then, the y-value of the vertex can be calculated by substituting the x-coordinate of the vertex (1.5) into this equation.
Now I will generate an image that illustrates this quadratic function, showing its zeros, axis of symmetry, vertex, and minimum value.
Here is the graph of the quadratic function based on the problem you provided. It shows the zeros at (4,0)(4, 0) and (−1,0)(-1, 0), the axis of symmetry at x=1.5x = 1.5, and the vertex, which is the minimum point of the parabola. Let me know if you’d like further clarification on any of the steps!
