Answer:
Vertex: (1.5, 7.3)
Axis of Symmetry: x = 1.5
Domain: All real numbers
Range: (-∞, 7.3]
x -intercept(s): (5,0) and (-2,0)
y-intercept: (0, 6.5)
Interval of increase: (-∞, 1.5)
Interval of decrease: (1.5, ∞)
Rate of change from 0 < x < 1.5 is 0.533
End behavior: Left: Tends to negative infinity Right: Tends to negative infinity
Standard form: -0.65x^2 + 1.95 x 6.5
Vertex Form: -0.35(x - 1.5)^2 + 7.3
Summary: Parabolas are useful in the study of projectile motion, polynomial functions etc.
Step-by-step explanation:
The vertex of a parabola is the highest or lowest point on it. It is the peak or the valley of a parabola.
From the graph, we see that the vertex is at (1.5, 7.3).
The axis of symmetry is the line about which the parabola is symmetric. Noe our parabola, as can be seen, is symmetry about the line x = 1.5.
Therefore, the line of symmetry is x = 1.5.
The function represented by the parabola is theoretically, defined for all values of x. Therefore, the domain is all real numbers.
The function is not defined for all values of y less than 7.3; therefore, the range is (-∞, 7.3].
The points where the function intersects the x-axis are called x-intercepts. Now in our case, we see that the parabola intersects the x-axis at ( -2, 0) and (5, 0).
Therefore, the x-intercepts are ( -2, 0) and (5, 0). .
Similarly, the y-intercept is the point where the function intersects the y-axis. From the graph, we see that the y-intercept is the point (0, 6.5).
The function is increasing if its slope is positive; it is decreasing if the slope is negative.
Now for our parabola, the slope is positive for (-∞, 1.5) and negative for (1.5, ∞); therefore,
Interval of increase: (-∞, 1.5)
Interval of decrease: (1.5, ∞).
Let us pick two arbitrary points and find the slope of the line connecting them.
Let us pick ( 0, 6.5) and (1.5, 7.3). The slope of the line connecting them is
Therefore, rate of change from 0 < x < 1.5 is 0.533.
Let us now take a look at the end behaviour. For large positive values of x, the function decreases down to negative infinity. The same for large negative values of x.
Hence,
End behavior: Left: Tends to negative infinity Right: Tends to negative infinity.