Answer:
- vertex: (5, 3)
- axis of symmetry: x = 5
- direction of opening: downward
- max: y = 3
- y-intercept: -47
Explanation:
Most of the questions can be answered by comparing the given equation to vertex form.
Vertex form
The vertex form of the equation for a parabola is ...
y = a(x -h)² +k
where 'a' is a vertical scale factor, and (h, k) is the vertex. The sign of 'a' determines whether the parabola opens upward or downward.
Parameter values
When we compare the given equation to the vertex form, we can easily see the values of 'a' and (h, k).
y = -2(x -5)² +3 . . . . . . . given equation
y = a(x -h)² +k . . . . . . . vertex form
The parameters are ...
a = -2, (h, k) = (5, 3)
We notice ...
- the sign of 'a' is negative
- the vertex is (5, 3)
Axis of symmetry
The axis of symmetry is the vertical line through the vertex. It has the equation ...
x = h
x = 5 . . . . equation of the axis of symmetry
Direction of opening, max
The sign of 'a' is negative. This means that large x-values will result in large negative y-values. The parabola opens downward.
When the curve opens downward, it means the vertex is the highest point on the curve. The maximum is the y-value of the vertex: k. There is no minimum.
The maximum is y = 3.
Y-intercept
The y-intercept is where the graph crosses the y-axis. It can be found by setting x=0, and computing the value of y:
y = -2(0 -5)² +3
y = -2(25) +3 = -50 +3
y = -47 . . . . the y-intercept.
The ordered pair for the point there is (0, -47).
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We show the graph so you can see how these features relate to the shape of the graph and points on it.