Consider the function y equals 3 x squared plus 12 x minus 23.

Question

Consider the function y equals 3 x squared plus 12 x minus 23.
​(a) Give the coordinates of the vertex of the graph of the function.
​(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
​(a) The vertex of the graph of the function is at the point 
  
enter your response here. ​(Type an ordered​ pair.)
Part 2
​(b) Choose the correct graph below. Each graph shown below has a window of ​[minus​10, ​10] by ​[minus​40, ​10].
A.

A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum in the third quadrant and passes through the positive and negative x-axis.
B.

A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
C.

A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum on the negative y-axis and passes through the positive and negative x-axis.
D.

GPT-4o mini · Answer

To analyze the function $ y = 3x^2 + 12x - 23 $, we can find the vertex and graph it. ### Part 1: Finding the Vertex The vertex of a quadratic function represented in the form $ y = ax^2 + bx + c $ can be found using the formula: $$ x = -\frac{b}{2a} $$ For the given function: - $ a = 3 $ - $ b = 12 $ Plugging in the values: $$ x = -\frac{12}{2(3)} = -\frac{12}{6} = -2 $$ Now, we can substitute $ x = -2 $ back into the function to find the corresponding $ y $-coordinate: $$ y = 3(-2)^2 + 12(-2) - 23 $$ $$ y = 3(4) - 24 - 23 $$ $$ y = 12 - 24 - 23 = 12 - 47 = -35 $$ Thus, the vertex of the graph of the function is at the point $(-2, -35)$. ### Part 2: Select the Correct Graph Since the function has a positive coefficient for $ x^2 $ ($ a = 3 > 0 $), it will be a **U-shaped** parabola, which means the vertex is a **minimum** point. Given the coordinates of the vertex $(-2, -35)$, we check the description of the graph: - **Graph : Upward U-shaped curve, minimum in the third quadrant. (This fits our vertex being at $(-2, -35)$.) - **Graph B**: Downward U-shaped curve. (Does not fit.) - **Graph C**: Upward U-shaped curve, minimum on the negative y-axis. (This does not specify if it matches the vertex location.) - **Graph D**: (No description provided.) Based on the details, **Graph is most likely the correct graph since it matches the characteristics of the quadratic function and the vertex location. ### Final Answers **Part 1:** The vertex of the graph of the function is at the point $ (-2, -35) $. **Part 2:** The correct graph is **.