“EXAMPLE 1 Using Symmetry to Graph a Quadratic Function Graph f(x) =−2(x + 3)2 + 4. Label the vertex and axis of symmetry. SOLUTION Step 1 Identify the constants a =−2, h=−3, and k= 4. Step 2 Plot the vertex (h, k) = (−3, 4) and draw the axis of symmetry x =−3. Step 3 Evaluate the function for two values of x, such as x =−2 and x =−1. f(−2) =−2(−2 + 3)2 + 4 = 2 f(−1) =−2(−1 + 3)2 + 4 =−4 Plot the points (−2, 2), (−1, −4), and their refl ections in the axis of symmetry. Step 4 Draw a parabola through the plotted points. y (−3, 4) 4 2 −6 −2 x x =−3 Quadratic functions can also be written in standard form, f(x) = ax2 + bx + c, where a ≠ 0. You can derive the standard form by expanding the vertex form. f (x) = a(x− h)2 + k f (x) = a(x2− 2hx + h2) + k f (x) = ax2− 2ahx + ah2 + k f (x) = ax2 + (−2ah)x + (ah2 + k) Vertex form Expand (x − h)2 . Distributive Property Group like terms. f (x) = ax2 + bx + c Let b =−2ah and let c = ah2 + k. This allows you to make the following observations. a = a: So, a has the same meaning in vertex form and standard form. b=−2ah: b Solve for h to obtain h=− — 2a . So, the axis of symmetry b is x =− — 2a . c = ah2 + k: In vertex form f(x) = a(x− h)2 + k, notice that f(0) = ah2 + k. So, c is the y-intercept.”