Draw the graph of \( f(x)=9(x+4)^{2}+3 \)

To graph the function \( f(x) = 9(x + 4)^2 + 3 \), we can follow these steps: 1. **Identify the Vertex**: The function is in the vertex form of a quadratic equation, which is \( f(x) = a(x - h)^2 + k \). Here, \( a = 9 \), \( h = -4 \), and \( k = 3 \). Thus, the vertex of the parabola is at the point \( (-4, 3) \). 2. **Determine the Direction**: Since \( a = 9 \) is positive, the parabola opens upwards. 3. **Find the Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \( x = -4 \). 4. **Calculate Additional Points**: To get a better idea of the shape of the graph, we can calculate a few additional points by choosing values of \( x \) around the vertex. - For \( x = -5 \): \[ f(-5) = 9(-5 + 4)^2 + 3 = 9(-1)^2 + 3 = 9 + 3 = 12 \] So, the point is \( (-5, 12) \). - For \( x = -3 \): \[ f(-3) = 9(-3 + 4)^2 + 3 = 9(1)^2 + 3 = 9 + 3 = 12 \] So, the point is \( (-3, 12) \). - For \( x = -6 \): \[ f(-6) = 9(-6 + 4)^2 + 3 = 9(-2)^2 + 3 = 9(4) + 3 = 36 + 3 = 39 \] So, the point is \( (-6, 39) \). - For \( x = -2 \): \[ f(-2) = 9(-2 + 4)^2 + 3 = 9(2)^2 + 3 = 9(4) + 3 = 36 + 3 = 39 \] So, the point is \( (-2, 39) \). 5. **Plot the Points**: Now we can plot the points we have: - Vertex: \( (-4, 3) \) - Points: \( (-5, 12) \), \( (-3, 12) \), \( (-6, 39) \), \( (-2, 39) \) 6. **Draw the Parabola**: Connect the points with a smooth curve to form the parabola. ### Summary of Key Points: - Vertex: \( (-4, 3) \) - Opens upwards - Axis of symmetry: \( x = -4 \) - Additional points: \( (-5, 12) \), \( (-3, 12) \), \( (-6, 39) \), \( (-2, 39) \) This will give you a clear graph of the function \( f(x) = 9(x + 4)^2 + 3 \).