I choose Quadratic Functions because they were an interesting advanced version of normal function and graphs. Analyzing Quadratic function is important in order to understand its components and predict the trends of its graphed form. These are the steps in order to fully analyze a quadratic function.

1. First we must understand what a quadratic function is when in the form where a, b , and  c  are constants. The graph of a quadratic function is a parabola.

2. Identify the Key Features:
– Vertex: The vertex of a parabola is the highest or lowest point on the graph and can be found with -b being the x cordinate and c being the y .
– Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. It is given by finding the x cordinate of the vertex.

–  Slope: The a is how wider the parobola is with higher the number the thiner paraobola
– Intercepts The x-intercepts are the points where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis. To find the x-intercepts, set and solve for x . The y-intercept is .

3. Determine the Direction of Opening:
– If a > 0 , the parabola opens upward.
– If a < 0 , the parabola opens downward.

4. Plot Points and Sketch the Graph:
– Use the vertex, intercepts, and symmetry to plot points and sketch the graph of the quadratic function.

5. Analyze the Graph:
– Determine the domain and range of the function.
– Identify any maximum or minimum values.
– Check for symmetry and whether the function is increasing or decreasing.

Sure, let’s analyze the quadratic function using the steps we discussed earlier:

1. Understand the Function: The given quadratic function is in the form .

2. Identify Key Features:
– Vertex: The vertex form of a quadratic function is , where (h, k) is the vertex. In this case, the vertex is (-5, 4).
– Axis of Symmetry: The axis of symmetry is given by the equation \x = -5 .
– Intercepts: To find the x-intercepts, set. However, this function has no real roots because the square term is always non-negative. The y-intercept is .

3. Direction of Opening: Since the coefficient of the squared term is positive, the parabola opens upward.

4. Plot Points and Sketch the Graph:
– Plot the vertex (-5, 4) and the y-intercept (0, 29). Since there are no real x-intercepts, the graph does not intersect the x-axis.
– Use the symmetry to plot other points if needed.

5. Analysis:
– Domain: All real numbers.
– Range: (since it opens upward).
– Minimum: The vertex (-5, 4) is the minimum point.
– Increasing or Decreasing: The function increases as you move to the right from the vertex and decreases as you move to the left.