Answer:
- b
- b
Step-by-step explanation:
The vertex and direction of opening can be read from the vertex-form equation of a quadratic.
Direction of opening
Consider the "parent" quadratic function ...
y = x²
For this function, the value of y cannot be negative. Larger positive values of x will give larger positive values of y. And, negative values of x that have greater magnitude (are farther from the y-axis) will also give larger positive values of y.
This means that the farther away from the y-axis an x-value is, the farther away from the x-axis is the corresponding y-value. The graph of this is said to "open upward." This will be the case for any positive coefficient of x². (Red graph in the first attachment.)
If the coefficient of x² is negative, then larger-magnitude x-values result in more negative y-values. This makes the graph "open downward." (Blue graph in the first attachment.)
Vertex
The vertex form of a quadratic equation is ...
y = a(x -h)² +k . . . . . . vertex (h, k); vertical scale factor 'a'
The value of 'a' is the coefficient of the x² term when this is simplified to standard form: y = ax² +bx +c. That is, the sign of 'a' tells you whether the graph opens upward (a > 0) or downward (a < 0).
The other constants in the vertex form equation tell you how the function has been translated. The value of k is a vertical translation quantity. Since it is added to each function value, it tells the number of units the function is translated upward.
The value h is a horizontal translation quantity. It seems slightly counter-intuitive that the function graph is translated to the right h units when h is subtracted from the x-value. That is the case.
As you may have noticed from the graphs in the first attachment, the vertex (turning point) of the parent function graph is at (x, y) = (0, 0). The vertex of the function ...
y = a(x -h)² +k
is located at (x, y) = (h, k).
1.
You want the opening direction of y = -2(x +3)² -1. You need look no further than the leading minus sign. It tells you the graph opens downward. (Red graph in the second attachment.)
Of course, the -1 at the end of the equation is the vertical translation of the vertex. That vertex is (-3, -1), as shown by the graph.
2.
You want the vertex of y = -1/2(x +5)² +7. Writing this so the binomial term has a minus sign, we have ...
y = -1/2(x -(-5))² +7
Comparing this to the vertex form ...
y = a(x -h)² +k
we identify the parameters to be ...
a = -1/2 . . . . opens downward
h = -5
k = 7 . . . . the vertex is (h, k) = (-5, 7)
The graph is the blue graph in the second attachment. It opens downward and has its vertex at (-5, 7).
__
Additional comment
A lot of math is about matching patterns. Here, you're asked to match the given equations to the "vertex form" pattern, and identify corresponding parts of the pattern: the leading coefficient (a), the horizontal translation (h), and the vertical translation (k). Once you know what the parts of the pattern mean, you can answer the questions easily.
