47. The equation y = 4.5x² + 9x + 15 describes the height of a diver, in metres, at x seconds.
Change this equation into vertex form by completing the square. State the y-intercept, the equation of the axis of symmetry, and the vertex.
B. Before locating the x-intercepts of the parabola, state the number of roots of the equation.
C. Use the quadratic equation formula to determine the x-intercepts of the parabola when the diver hits the water.
D. Explain why one root is inadmissible. Make a sketch to represent the path of the diver.
y = 4.5x² + 9x + 15
4.5 x^2 + 9 x = y - 15
x^2 + 2 x = (10/45)(y-15)
x^2 + 2 x +(2/2)^2 = (10/45)(y-15) + 1
You have a sign error, the height goes to + infinity for large + or - x
I will lay it out your way but suspect
h = initial height = 15 m
Vi = initial speed up = 9 m/s
4.5 = 9.81/2 = g/2 = half gravity
so
h = Hi + Vi t - (g/2)t^2
is here
h = 15 + 9 t - 4.9 t^2
h = 0 is hit the water
One of your times will be negative, that was before you jumped :)
but anyway using what you have:
(x+1)^2 = (10/45)( y-15 ) + (10/45)(45/10)
(x+1)^2 = (10/45 ( y -15+45/10) = (10/45)(y -10.5)
vertex at (- 1, 10.5) symmetric about x = -1
for x axis crossings put y = 0 in
x+1 = /-(10/45) sqrt (-10.5), imaginary, always above axis of course
when x = 0, y axis intercept = 15
now do it right with gravity down :)
To change the given equation into vertex form by completing the square, follow these steps:
Step 1: Write the equation in the form y = ax² + bx + c.
Given equation: y = 4.5x² + 9x + 15
Step 2: Group the x-terms and the constant term separately.
y = (4.5x² + 9x) + 15
Step 3: Factor out the coefficient of x² from the x-terms.
y = 4.5(x² + 2x) + 15
Step 4: To complete the square, take half of the coefficient of x, square it, then add and subtract it inside the parentheses.
y = 4.5(x² + 2x + 1 - 1) + 15
Step 5: Rearrange the equation.
y = 4.5((x + 1)² - 1) + 15
Step 6: Expand the equation.
y = 4.5(x + 1)² - 4.5 + 15
Step 7: Simplify.
y = 4.5(x + 1)² + 10.5
Now, the equation is in vertex form, y = a(x - h)² + k, where (h, k) represents the vertex.
From the equation above, the y-intercept is found by setting x = 0.
y = 4.5(0 + 1)² + 10.5
y = 4.5 + 10.5
y = 15
So, the y-intercept is 15.
The equation of the axis of symmetry is x = -h.
In this case, the value of h is -1.
So, the equation of the axis of symmetry is x = -(-1), which simplifies to x = 1.
To find the vertex, set the x-term equal to 0.
(x + 1)² = 0
x + 1 = 0
x = -1
Therefore, the vertex is (-1, 10.5).
B. To locate the x-intercepts (roots) of the parabola, we need to determine the number of roots of the equation.
The discriminant (b² - 4ac) can be used to determine the number of roots. If the discriminant is greater than zero, the equation has two distinct real roots. If the discriminant is equal to zero, the equation has one real root (called a repeated root or double root). If the discriminant is less than zero, the equation has no real roots.
For the given equation y = 4.5x² + 9x + 15, we can find the discriminant using the formula.
Discriminant (D) = b² - 4ac
Plugging in the values from the equation, we have:
D = (9)² - 4(4.5)(15)
D = 81 - 270
D = -189
Since the discriminant (-189) is less than zero, the equation has no real roots. Therefore, there are no x-intercepts.
C. Since there are no x-intercepts, the quadratic equation formula cannot be used to determine the x-intercepts of the parabola when the diver hits the water.
D. One root is inadmissible because the height of the diver cannot be negative. The parabola opens upward, indicating that the height is increasing over time. Therefore, any negative root would be considered inadmissible in this context.
To represent the path of the diver, you can sketch a graph of the quadratic equation y = 4.5x² + 9x + 15. The vertex of the parabola represents the highest point the diver reaches. The parabola opens upward, showing the diver's upward motion before reaching the maximum height and then descending.