1Cademy - Graphing $$y = x^2 + 4x + 5$$

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Graphing a Quadratic Equation in Two Variables

Example

Graphing $y = x^2 + 4x + 5$

To graph the parabola $y = x^2 + 4x + 5$ , follow the systematic graphing procedure:

Step 1: The coefficient $a = 1$ is positive, so the parabola opens upward.

Step 2: Find the axis of symmetry using $x = -\frac{b}{2a}$ . With $a = 1$ and $b = 4$ , $x = -\frac{4}{2(1)} = -2$ . The axis of symmetry is the vertical line $x = -2$ .

Step 3: Find the vertex by substituting $x = -2$ into the equation: $y = (-2)^2 + 4(-2) + 5 = 4 - 8 + 5 = 1$ . The vertex is the point $(-2, 1)$ .

Step 4: Find the $y$ -intercept by setting $x = 0$ : $y = 0^2 + 4(0) + 5 = 5$ . The $y$ -intercept is $(0, 5)$ . The point symmetric to the $y$ -intercept across the axis of symmetry $x = -2$ is $(-4, 5)$ .

Step 5: Find the $x$ -intercepts by setting $y = 0$ , yielding $0 = x^2 + 4x + 5$ . Testing the discriminant gives $b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4$ . Because the discriminant is negative, there are no real solutions, meaning the parabola has no $x$ -intercepts.

Step 6: Graph the parabola by plotting the vertex, the $y$ -intercept, and the symmetric point, then connecting them with a smooth, upward-opening curve.

Updated 2026-04-21

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OpenStax Elementary Algebra

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