This answer key provides solutions for a typical Algebra 1 worksheet on graphing quadratic functions. Remember that specific questions on your worksheet may vary slightly, but the underlying principles remain the same. Always refer to your specific worksheet for the exact questions and context.
Understanding Quadratic Functions
Before diving into the answers, let's quickly review the key components of graphing quadratic functions:
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Standard Form: A quadratic function is typically written in standard form:
f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. The value of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0). -
Vertex: The vertex is the lowest (minimum) or highest (maximum) point on the parabola. Its x-coordinate is given by
x = -b / 2a. Substitute this x-value back into the function to find the y-coordinate. -
Axis of Symmetry: This is a vertical line that passes through the vertex. Its equation is
x = -b / 2a. -
x-intercepts (Roots or Zeros): These are the points where the parabola intersects the x-axis (where y = 0). They can be found by solving the quadratic equation
ax² + bx + c = 0using factoring, the quadratic formula, or completing the square. -
y-intercept: This is the point where the parabola intersects the y-axis (where x = 0). It's simply the value of 'c' in the standard form.
Sample Worksheet Problems and Solutions
Let's consider a few example problems that might appear on your worksheet:
Problem 1: Graph the quadratic function f(x) = x² + 4x + 3
Solution:
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Identify a, b, and c: a = 1, b = 4, c = 3. Since a > 0, the parabola opens upwards.
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Find the vertex:
- x-coordinate: x = -b / 2a = -4 / (2 * 1) = -2
- y-coordinate: f(-2) = (-2)² + 4(-2) + 3 = -1
- Vertex: (-2, -1)
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Find the axis of symmetry: x = -2
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Find the y-intercept: The y-intercept is (0, 3) (since c = 3).
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Find the x-intercepts: Solve x² + 4x + 3 = 0. This factors to (x + 1)(x + 3) = 0, giving x-intercepts (-1, 0) and (-3, 0).
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Graph: Plot the vertex, axis of symmetry, y-intercept, and x-intercepts. Sketch a smooth parabola through these points.
Problem 2: Find the vertex and axis of symmetry for the quadratic function g(x) = -2x² + 8x - 6
Solution:
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Identify a, b, and c: a = -2, b = 8, c = -6. Since a < 0, the parabola opens downwards.
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Find the vertex:
- x-coordinate: x = -b / 2a = -8 / (2 * -2) = 2
- y-coordinate: g(2) = -2(2)² + 8(2) - 6 = 2
- Vertex: (2, 2)
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Find the axis of symmetry: x = 2
Problem 3: Determine whether the parabola represented by the equation h(x) = 3x² - 5 opens upwards or downwards.
Solution: Since a = 3 (positive), the parabola opens upwards.
Remember: This is a general guide. Your worksheet might include additional problem types, such as completing the square to find the vertex form, or using the quadratic formula to find x-intercepts when factoring is not easy. Make sure to carefully review your specific problems and apply the relevant concepts. If you're still struggling with specific problems, consult your textbook or teacher for further assistance.
