Graph the parabola (do not place any dots) then fill in the blanks about the vertex, any intercepts, the domain, and range. Enter intercepts as ordered pairs, aka points. \( y=-x^{2}+4 x \)

To graph the parabola given by the equation \( y = -x^2 + 4x \), we can start by rewriting it in vertex form and identifying key features such as the vertex, intercepts, domain, and range. ### Step 1: Rewrite in Vertex Form The given equation is \( y = -x^2 + 4x \). We can complete the square to rewrite it in vertex form. 1. Factor out the negative sign from the quadratic terms: \[ y = -(x^2 - 4x) \] 2. Complete the square inside the parentheses: - Take half of the coefficient of \( x \) (which is -4), square it, and add/subtract it inside the parentheses: \[ y = -\left(x^2 - 4x + 4 - 4\right) = -\left((x - 2)^2 - 4\right) \] \[ y = - (x - 2)^2 + 4 \] Now, the equation is in vertex form \( y = - (x - 2)^2 + 4 \). ### Step 2: Identify the Vertex From the vertex form \( y = a(x - h)^2 + k \), we can see that the vertex \( (h, k) \) is: - Vertex: \( (2, 4) \) ### Step 3: Find the Intercepts **Y-Intercept:** To find the y-intercept, set \( x = 0 \): \[ y = -0^2 + 4(0) = 0 \] - Y-Intercept: \( (0, 0) \) **X-Intercepts:** To find the x-intercepts, set \( y = 0 \): \[ 0 = -x^2 + 4x \] Factoring gives: \[ 0 = x(-x + 4) \] So, \( x = 0 \) or \( x = 4 \). - X-Intercepts: \( (0, 0) \) and \( (4, 0) \) ### Step 4: Determine the Domain and Range **Domain:** The domain of a parabola is all real numbers: - Domain: \( (-\infty, \infty) \) **Range:** Since the parabola opens downwards (as indicated by the negative coefficient of the \( x^2 \) term), the range is all values less than or equal to the y-coordinate of the vertex: - Range: \( (-\infty, 4] \) ### Summary - **Vertex:** \( (2, 4) \) - **X-Intercepts:** \( (0, 0) \) and \( (4, 0) \) - **Y-Intercept:** \( (0, 0) \) - **Domain:** \( (-\infty, \infty) \) - **Range:** \( (-\infty, 4] \) This completes the analysis of the parabola \( y = -x^2 + 4x \).