Question

quadratic functions and equations
comparing properties of quadratic functions given in different forms
function 1
function 2
f(x) = -3x² -12x -7
(a) what is the vertex of function 1?
(b) what is the vertex of function 2?
(c) which function has the larger maximum value?
function 1
function 2
what is the larger maximum value?

Explanation:

Part (a) - Vertex of Function 1

Step 1: Identify the vertex from the graph

The graph of Function 1 is a parabola. The vertex is the highest point (since it opens downward) of the parabola. From the graph, we can see that the vertex is at the point where \( x = 2 \) and \( y = 3 \) (by looking at the coordinates of the peak of the parabola). Wait, let's check the grid. The x - axis: the vertex is at \( x = 2 \)? Wait, no, looking at the graph, the vertex is at (2, 3)? Wait, no, let's look again. The graph has points: when x = 0, y = 1? Wait, no, the blue points: the vertex is at (2, 3)? Wait, no, the graph's vertex is at (2, 3)? Wait, the graph of Function 1: the peak is at x = 2, y = 3? Wait, no, let's check the coordinates. The x - coordinate of the vertex: the parabola is symmetric. The roots (where y = 0) seem to be at x = - 1 and x = 5? Wait, no, the blue points on the x - axis: one at x = 0 (y = 1?) No, the graph: the vertex is at (2, 3). Wait, maybe I made a mistake. Wait, the graph: the vertex is the highest point. Let's see the x - coordinate: the parabola is symmetric. The distance between the two x - intercepts (where y = 0) is from x = - 1 to x = 5? Wait, no, the blue points on the x - axis: one at x = 0 (y = 1) no, the graph: the vertex is at (2, 3). Wait, maybe the correct vertex for Function 1 is (2, 3). Wait, no, let's look at the grid. The y - axis: the vertex is at y = 3, x = 2. So the vertex of Function 1 is (2, 3).

Part (b) - Vertex of Function 2

Step 1: Recall the formula for the vertex of a quadratic function

For a quadratic function in the form \( f(x)=ax^{2}+bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). The function is \( f(x)=-3x^{2}-12x - 7 \), so \( a=-3 \), \( b = - 12 \), \( c=-7 \).

Step 2: Calculate the x - coordinate of the vertex

Using the formula \( x =-\frac{b}{2a} \), substitute \( a=-3 \) and \( b=-12 \):
\( x=-\frac{-12}{2\times(-3)}=-\frac{-12}{-6}=- 2 \)

Step 3: Calculate the y - coordinate of the vertex

Substitute \( x = - 2 \) into the function \( f(x)=-3x^{2}-12x - 7 \):
\( f(-2)=-3\times(-2)^{2}-12\times(-2)-7=-3\times4 + 24-7=-12 + 24-7 = 5 \)
So the vertex of Function 2 is (-2, 5).

Part (c) - Larger Maximum Value

Step 1: Find the maximum value of each function

• For Function 1, the vertex is (2, 3), so the maximum value (since it opens downward) is 3.
• For Function 2, the vertex is (-2, 5), so the maximum value (since \( a=-3<0 \), it opens downward) is 5.

Step 2: Compare the maximum values

Since \( 5>3 \), Function 2 has the larger maximum value, and the larger maximum value is 5.

Answer:

(a) The vertex of Function 1 is \((2, 3)\)
(b) The vertex of Function 2 is \((-2, 5)\)
(c) The function with the larger maximum value is Function 2, and the larger maximum value is \(5\)