Question

graph the equation.
y = -1(x - 3)^2 + 4

Explanation:

Step1: Identify the vertex form

The equation \( y = -1(x - 3)^2 + 4 \) is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( h = 3 \), \( k = 4 \), so the vertex is \((3, 4)\).

Step2: Determine the direction of opening

Since \( a = -1 \) (negative), the parabola opens downward.

Step3: Find the axis of symmetry

The axis of symmetry is the vertical line \( x = h \), so \( x = 3 \).

Step4: Plot the vertex and use symmetry

Plot the vertex \((3, 4)\). Then, choose a value for \( x \) (e.g., \( x = 2 \)): \( y = -1(2 - 3)^2 + 4 = -1(1) + 4 = 3 \), so the point \((2, 3)\) is on the parabola. By symmetry, the point \((4, 3)\) (since \( x = 4 \) is 1 unit from \( x = 3 \) as \( x = 2 \)) is also on the parabola. Similarly, for \( x = 1 \): \( y = -1(1 - 3)^2 + 4 = -4 + 4 = 0 \), so \((1, 0)\) and \((5, 0)\) (symmetric) are on the parabola. Connect these points to form the downward - opening parabola with vertex \((3, 4)\).

Answer:

The graph is a parabola with vertex at \((3, 4)\), opening downward, and axis of symmetry \( x = 3 \). (To draw it, plot the vertex, use symmetry to find other points, and connect them.)