In a quadratic function of the form $f(x) = ax^2 + bx + c$, the coefficients $a$, $b$, and $c$ play crucial roles in determining the characteristics of the parabola. Here’s how each coefficient influences the function:

1. Coefficient $a$ (Shape and Direction):

   • Shape: The coefficient $a$ affects the "width" or "steepness" of the parabola. If $|a|$ (the absolute value of $a$) is large, the parabola is narrower, and if $|a|$ is small, the parabola is wider.
   • Direction: The sign of $a$ determines whether the parabola opens upwards or downwards. If $a > 0$, the parabola opens upwards (convex), and if $a < 0$, the parabola opens downwards (concave).

2. Coefficient $b$ (Position and Axis of Symmetry):

   • The coefficient $b$ influences the position of the vertex horizontally along the x-axis.
   • The axis of symmetry of the parabola is given by the line $x = \frac{-b}{2a}$. This is the vertical line that passes through the vertex of the parabola and divides it into two symmetric halves.

3. Coefficient $c$ (Vertical Position):

   • The coefficient $c$ represents the y-intercept of the parabola, which is the point where the graph intersects the y-axis. This is because $f(0) = c$.
   • It gives the vertical position of the parabola on the graph.

Steps to Analyze a Quadratic Function:

1. Identify the coefficients: Given $f(x) = ax^2 + bx + c$, identify the values of $a$, $b$, and $c$.
2. Determine the direction: Check the sign of $a$ to know whether the parabola opens upwards ( $a > 0$ ) or downwards ( $a < 0$ ).
3. Find the axis of symmetry: Calculate the axis of symmetry using $x = \frac{-b}{2a}$.
4. Locate the vertex: Use the axis of symmetry to find the x-coordinate of the vertex, and then plug it into the function to find the y-coordinate.
5. Identify the y-intercept: The value of $c$ directly gives the y-intercept.

Example:

Consider the quadratic function $f(x) = 2x^2 - 4x + 1$.

• Identify coefficients: $a = 2$, $b = -4$, $c = 1$.
• Direction: $a = 2$ (positive), so the parabola opens upwards.
• Axis of symmetry: $x = \frac{-(-4)}{2(2)} = \frac{4}{4} = 1$.
• Vertex: Substitute $x = 1$ into $f(x)$ to find the y-coordinate: $f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1$. So, the vertex is $(1, -1)$.
• Y-intercept: $c = 1$, so the parabola intersects the y-axis at $(0, 1)$.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How can the quadratic formula be derived from the standard form of a quadratic equation?
2. Can you explain the process of completing the square to find the vertex form of a quadratic function?
3. What are the applications of quadratic functions in physics and engineering?
4. How do you graph a quadratic function step-by-step?
5. What are the implications of a negative discriminant in a quadratic equation?

Tip: Always graph the quadratic function after analyzing the coefficients to visualize the shape, direction, and key points such as the vertex and intercepts.