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. First, we'll identify the vertex. For a quadratic function in the form ax squared plus bx plus c, the x-coordinate of the vertex is negative b divided by 2a. In our function, a equals negative 1 and b equals 4, so the x-coordinate is negative 4 divided by negative 2, which equals 2. Substituting x equals 2 into the function, we get f of 2 equals negative 4 plus 8 plus 5, which equals 9. So the vertex is at the point (2, 9). The axis of symmetry passes through the vertex, so it's the vertical line x equals 2. Now, let's find the intercepts of our quadratic function. For the x-intercepts, we set f of x equals zero and solve for x. We have negative x squared plus 4x plus 5 equals 0. Multiplying by negative 1, we get x squared minus 4x minus 5 equals 0. Factoring this expression, we get (x minus 5) times (x plus 1) equals 0. Setting each factor equal to zero, we get x equals 5 or x equals negative 1. So our x-intercepts are the points (-1, 0) and (5, 0). For the y-intercept, we find f of 0, which equals negative 0 squared plus 4 times 0 plus 5, which equals 5. So our y-intercept is the point (0, 5). Let's summarize all the key features of our quadratic function f of x equals negative x squared plus 4x plus 5. The vertex is at the point (2, 9), which is the maximum point of the parabola since the coefficient of x squared is negative. The axis of symmetry is the vertical line x equals 2, which passes through the vertex. The x-intercepts, where the graph crosses the x-axis, are at (-1, 0) and (5, 0). The y-intercept, where the graph crosses the y-axis, is at (0, 5). The domain of this function is all real numbers, or negative infinity to positive infinity, since a quadratic function is defined for all x values. The range is all y values less than or equal to 9, or negative infinity to 9, because the parabola opens downward and has its maximum at y equals 9. The parabola opens downward because the coefficient of x squared is negative. Now, let's convert our quadratic function from standard form to vertex form. The standard form is f of x equals negative x squared plus 4x plus 5. To convert to vertex form, we complete the square for the x terms. First, we group the x terms: negative x squared plus 4x. The coefficient of x is 4, so half of that is 2, and 2 squared is 4. We add and subtract 4 to maintain equality: f of x equals negative x squared plus 4x minus 4 plus 5 plus 4, which simplifies to negative x squared plus 4x minus 4 plus 9. We can factor the first three terms: negative times (x squared minus 4x plus 4) plus 9, which gives us negative times (x minus 2) squared plus 9. This is our vertex form: f of x equals negative (x minus 2) squared plus 9. From this form, we can directly read that the vertex is at (2, 9), confirming our earlier calculation. The vertex form clearly shows that this parabola is a transformation of the basic parabola y equals x squared. It's reflected across the x-axis because of the negative coefficient, shifted 2 units right, and shifted 9 units up. To summarize what we've learned about the quadratic function f of x equals negative x squared plus 4x plus 5: The vertex is at the point (2, 9), which is the maximum point of the parabola since it opens downward. The axis of symmetry is the vertical line x equals 2. The x-intercepts, where the graph crosses the x-axis, are at negative 1, 0 and 5, 0, while the y-intercept is at 0, 5. In vertex form, the function can be written as f of x equals negative (x minus 2) squared plus 9, which clearly shows that the basic parabola y equals x squared has been reflected across the x-axis, shifted 2 units right, and shifted 9 units up. These key features provide a complete understanding of the graph's behavior and position.