Let’s analyze the quadratic function

y = x² + 4x – 5.

Below are several useful forms and properties of this quadratic.

1. Factored Form:
  We seek two numbers that multiply to –5 and add to 4. These numbers are 5 and –1.
  Thus, we can factor the expression as:
    x² + 4x – 5 = (x + 5)(x – 1).

2. Zeros (x-intercepts):
  Set y = 0:
    (x + 5)(x – 1) = 0.
  So, the zeros are:
    x + 5 = 0 ⟹ x = –5,
    x – 1 = 0 ⟹ x = 1.

3. Vertex (and Completing the Square):
  (a) Using the formula, the x-coordinate of the vertex is given by
    x = –b/(2a) = –4/(2·1) = –2.
  (b) Substitute x = –2 into the function:
    y = (–2)² + 4(–2) – 5 = 4 – 8 – 5 = –9.
  Thus, the vertex is at (–2, –9).

Alternatively, complete the square:

y = x² + 4x – 5
    = (x² + 4x + 4) – 4 – 5    [adding and subtracting 4]
    = (x + 2)² – 9.

This is the vertex form, which confirms that the vertex is (–2, –9).

4. Axis of Symmetry:
  The axis of symmetry is the vertical line through the vertex:
    x = –2.

5. Additional Information:
  • The y-intercept occurs when x = 0:
    y = 0² + 4·0 – 5 = –5,
    so the y-intercept is (0, –5).
  • Since the coefficient of x² is positive, the parabola opens upward, and its minimum value is y = –9 at the vertex.

Summary:
  • Standard Form: y = x² + 4x – 5
  • Factored Form: y = (x + 5)(x – 1)
  • Vertex Form:  y = (x + 2)² – 9
  • Vertex:    (–2, –9)
  • Zeros:    x = –5 and x = 1
  • Axis of Symmetry: x = –2
  • y-intercept:  (0, –5)

This fully characterizes the quadratic function given in problem 3.2.1.

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Question

$3.2.1 $ y=x^{2}+4 x-5 $$

3.2.1 $ y=x^{2}+4 x-5 $

Ask by May Williams.

Feb 16,2025 11:46

UpStudy AI Solution

Tutor-Verified Answer

Answer

The quadratic function $ y = x^{2} + 4x - 5 $ has the following properties: - **Zeros**: $ x = -5 $ and $ x = 1 $ - **Vertex**: $ (-2, -9) $ - **Axis of Symmetry**: $ x = -2 $ - **Y-intercept**: $ (0, -5) $ - **Factored Form**: $ (x + 5)(x - 1) $ - **Vertex Form**: $ (x + 2)^2 - 9 $ The parabola opens upward with its minimum point at the vertex.

Solution

Let’s analyze the quadratic function y = x² + 4x – 5. Below are several useful forms and properties of this quadratic. 1. Factored Form: We seek two numbers that multiply to –5 and add to 4. These numbers are 5 and –1. Thus, we can factor the expression as: x² + 4x – 5 = (x + 5)(x – 1). 2. Zeros (x-intercepts): Set y = 0: (x + 5)(x – 1) = 0. So, the zeros are: x + 5 = 0 ⟹ x = –5, x – 1 = 0 ⟹ x = 1. 3. Vertex (and Completing the Square): (a) Using the formula, the x-coordinate of the vertex is given by x = –b/(2a) = –4/(2·1) = –2. (b) Substitute x = –2 into the function: y = (–2)² + 4(–2) – 5 = 4 – 8 – 5 = –9. Thus, the vertex is at (–2, –9). Alternatively, complete the square: y = x² + 4x – 5 = (x² + 4x + 4) – 4 – 5 [adding and subtracting 4] = (x + 2)² – 9. This is the vertex form, which confirms that the vertex is (–2, –9). 4. Axis of Symmetry: The axis of symmetry is the vertical line through the vertex: x = –2. 5. Additional Information: • The y-intercept occurs when x = 0: y = 0² + 4·0 – 5 = –5, so the y-intercept is (0, –5). • Since the coefficient of x² is positive, the parabola opens upward, and its minimum value is y = –9 at the vertex. Summary: • Standard Form: y = x² + 4x – 5 • Factored Form: y = (x + 5)(x – 1) • Vertex Form: y = (x + 2)² – 9 • Vertex: (–2, –9) • Zeros: x = –5 and x = 1 • Axis of Symmetry: x = –2 • y-intercept: (0, –5) This fully characterizes the quadratic function given in problem 3.2.1.

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Bonus Knowledge

This quadratic equation represents a parabola that opens upwards. Understanding its vertex and x-intercepts can be quite fun! The vertex can be found using the formula $ x = -\frac{b}{2a} $, which in this case gives $ x = -2 $. Plugging that back into the equation yields the minimum point of the parabola, adding a delightful touch to your graph! To find the x-intercepts, you can factor the equation or use the quadratic formula. Factoring $ y = (x + 5)(x - 1) $ shows where the parabola crosses the x-axis, giving you the intercepts at $ x = -5 $ and $ x = 1 $. That means the parabola has a lovely dance with the x-axis!