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 four x plus five. We need to find its vertex, axis of symmetry, x-intercepts, and y-intercept. This parabola opens downward since the coefficient of x squared is negative. To find the vertex, we use the formula h equals negative b over two a. For our function, a equals negative one and b equals four. So h equals negative four over two times negative one, which equals two. The y-coordinate k equals f of two, which is negative four plus eight plus five, equals nine. Therefore, the vertex is at the point two comma nine. The axis of symmetry is the vertical line x equals two, passing through the vertex. To find x-intercepts, we set the function equal to zero and solve negative x squared plus four x plus five equals zero. Factoring gives us x plus one times x minus five equals zero, so x equals negative one and x equals five. The y-intercept is found by evaluating f of zero, which equals five. Here's our complete analysis of f of x equals negative x squared plus four x plus five. The vertex is at two comma nine, which is the highest point since the parabola opens downward. The axis of symmetry is the line x equals two. The parabola crosses the x-axis at negative one comma zero and five comma zero, and crosses the y-axis at zero comma five. This gives us a complete picture of the quadratic function's behavior. To summarize what we've learned: The vertex formula gives us both the axis of symmetry and maximum point. For downward parabolas, the vertex represents the highest value. We find x-intercepts by solving the equation, and the y-intercept is simply the constant term. These key features together allow us to completely understand and sketch any quadratic function.