find the x and y intercepts
f(x)=9x^2+24x+16
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let y = 9x^2 + 24x + 16
for the y-intercept , let x = 0 , --->y = 16
for the x-intercept, let y = 0
9x^2 + 24x + 16 = 0
you could use the quadratic formula, but I noticed the above is a perfect square, so
(3x+4)^2 = 0
3x+4 = 0
x = -4/3
So there is one x-intercept at x = -4/3
(the parabola has its vertex at (-4/3, 0) and crosses the y-axis at (0,16) )
This crosses the y axis when x=0 so
y=16
This crosses the x axis when y=0
0=9x^2+24x+16
0=(x+4/3)^2
so x=-4/3
To find the x-intercepts, we set y (or f(x)) equal to zero and solve for x.
So, we have:
0 = 9x^2 + 24x + 16
To solve this quadratic equation, we can either use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation above, a = 9, b = 24, and c = 16. Applying the values to the formula:
x = (-24 ± √(24^2 - 4 * 9 * 16)) / (2 * 9)
Simplifying further:
x = (-24 ± √(576 - 576)) / 18
x = (-24 ± √0) / 18
x = (-24 ± 0) / 18
x = -24 / 18
x = -4/3
Therefore, the x-intercept for the equation f(x) = 9x^2 + 24x + 16 is -4/3.
To find the y-intercept, we substitute x = 0 into the equation.
f(0) = 9(0)^2 + 24(0) + 16
f(0) = 0 + 0 + 16
f(0) = 16
Therefore, the y-intercept is 16.
To find the x-intercepts of a function, we set the y-value equal to zero (f(x) = 0) and solve for x. Similarly, to find the y-intercept, we set the x-value equal to zero (x = 0) and solve for y.
Let’s find the x-intercepts first. For a quadratic equation in the form of f(x) = ax^2 + bx + c, the x-intercepts occur when the function equals zero, so we can set f(x) = 9x^2 + 24x + 16 equal to zero:
9x^2 + 24x + 16 = 0
Next, we can either factor the equation or use the quadratic formula to solve for x. In this case, factoring might not be easy, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 9, b = 24, and c = 16:
x = (-24 ± √(24^2 - 4 * 9 * 16)) / (2 * 9)
Simplifying further:
x = (-24 ± √(576 - 576)) / 18
x = (-24 ± √0) / 18
Since the discriminant (√(b^2 - 4ac)) is zero, it means there is only one real root or x-intercept for this quadratic equation. Let's simplify it:
x = -24/18
Simplifying further:
x = -4/3
Therefore, the quadratic function f(x) = 9x^2 + 24x + 16 has a single x-intercept at x = -4/3.
Now, let's find the y-intercept by setting x = 0:
f(0) = 9(0)^2 + 24(0) + 16
f(0) = 16
So, the y-intercept is at y = 16.
To summarize:
- The x-intercept is at (-4/3, 0)
- The y-intercept is at (0, 16)