key features of quadratics worksheet

2 min read 02-02-2025

This worksheet guide delves into the key features of quadratic functions, providing a structured approach to understanding their graphs and equations. We'll explore parabolas, vertices, intercepts, and more, equipping you with the tools to confidently analyze and solve quadratic problems.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve. The shape and position of the parabola are determined by the values of a, b, and c.

1. The Parabola's Orientation:

The value of a dictates whether the parabola opens upwards or downwards:

a > 0: The parabola opens upwards (a "smile"). This indicates a minimum value.
a < 0: The parabola opens downwards (a "frown"). This indicates a maximum value.

2. The Vertex:

The vertex is the turning point of the parabola. It represents either the minimum or maximum value of the function. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

Substitute this x-value back into the quadratic equation to find the y-coordinate of the vertex.

3. The Axis of Symmetry:

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is:

x = -b / 2a (Note: This is the same as the x-coordinate of the vertex).

4. x-intercepts (Roots or Zeros):

The x-intercepts are the points where the parabola intersects the x-axis (where y = 0). These are also known as the roots or zeros of the quadratic function. They can be found by setting f(x) = 0 and solving the quadratic equation using methods such as factoring, the quadratic formula, or completing the square.

5. y-intercept:

The y-intercept is the point where the parabola intersects the y-axis (where x = 0). It can be easily found by substituting x = 0 into the quadratic equation: y = c.

Worksheet Exercises:

Now, let's put this knowledge into practice with some exercises:

Exercise 1: For the quadratic function f(x) = 2x² - 8x + 6:

Determine if the parabola opens upwards or downwards.
Find the coordinates of the vertex.
Find the equation of the axis of symmetry.
Find the x-intercepts (roots).
Find the y-intercept.

Exercise 2: Sketch the graph of the quadratic function f(x) = -x² + 4x - 3, labeling the vertex, axis of symmetry, x-intercepts, and y-intercept.

Exercise 3: A ball is thrown upwards and its height (in meters) after t seconds is given by the equation h(t) = -5t² + 20t + 5.

What is the maximum height reached by the ball?
After how many seconds does the ball reach its maximum height?
When does the ball hit the ground?

Exercise 4: Find a quadratic function whose vertex is (2, 1) and passes through the point (0, 5).

These exercises cover various aspects of quadratic functions, allowing for a comprehensive understanding of their key features. Remember to show your working for each problem. Good luck!

This worksheet provides a solid foundation for understanding quadratic functions. Further exploration into topics like completing the square, the discriminant, and applications of quadratic equations will enhance your skills further.