f(x) = -x² + 4x + 5. Show the vertex, axis of symmetry, x-intercepts, and y- intercept

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Let's analyze the quadratic function f of x equals negative x squared plus 4x plus 5. The coefficients are a equals negative 1, b equals 4, and c equals 5. Since a is negative, the parabola opens downward with a maximum point at the vertex. To find the vertex, we use the formula h equals negative b over 2a. With b equals 4 and a equals negative 1, we get h equals 2. Then we substitute x equals 2 into the function to find k equals 9. The vertex is at point 2 comma 9, which is the highest point of this downward-opening parabola. The axis of symmetry is the vertical line x equals 2. To find the x-intercepts, we set f of x equal to zero. This gives us negative x squared plus 4x plus 5 equals zero. Multiplying by negative 1, we get x squared minus 4x minus 5 equals zero. We can factor this as x minus 5 times x plus 1 equals zero. This gives us x equals 5 and x equals negative 1. So the x-intercepts are at points negative 1 comma 0 and 5 comma 0. To find the y-intercept, we set x equal to zero and evaluate the function. f of 0 equals negative 0 squared plus 4 times 0 plus 5, which simplifies to 5. So the y-intercept is at point 0 comma 5. Now we have all the key features: the vertex at 2 comma 9, the axis of symmetry at x equals 2, the x-intercepts at negative 1 comma 0 and 5 comma 0, and the y-intercept at 0 comma 5. To summarize our analysis of f of x equals negative x squared plus 4x plus 5: The vertex is at 2 comma 9, which is the maximum point. The axis of symmetry is x equals 2. The parabola crosses the x-axis at negative 1 comma 0 and 5 comma 0, and crosses the y-axis at 0 comma 5. Since the coefficient of x squared is negative, the parabola opens downward.