Express f(x) = -x-4x+ 12 in the vertex form, fix) = ax + h)²+ k, and in the intercept form, flx)= a(x + p)(x +q). where a, h, k, p and q are constants. Hence, sketch the graph..
You really should proofread what you post. I'll go with
-x^2 - 4x + 12
-(x^2 + 4x) + 12
-(x^2 + 4x + 4) + 12 + 4
-(x+2)^2 + 16
-x^2 - 4x + 12
-(x^2 + 4x - 12)
-(x+6)(x-2)
There are lots of handy online graphing sites if you need help with that.
If your question means:
Express f(x) = - x² - 4 x + 12
then
add and subtract 4 to left side
f(x) = - x² - 4 x + 12 + 4 - 4
f(x) = - x² - 4 x - 4 + 12 + 4
f(x) = - ( x² + 4 x + 4 ) + 16
f(x) = - ( x² + 2 ∙ x ∙ 2 + 2² ) + 16
Since:
( a + b )² = a² + 2 ∙ a ∙ b + b²
( x + 2 )² = x² + 2 ∙ x ∙ 2 + 2²
f(x) = - ( x + 2 )² + 16
f(x) = a ( x - p ) ( x - q )
where p and q are the roots:
In this case:
a = - 1
- ( x + 2 )² + 16 = 0
Subtract 16 to both sides
- ( x + 2 )² = - 16
Multiply both sides by - 1
( x + 2 )² = 16
x + 2 = ± √16
x + 2 = ± 4
Subtract 2 to both sides
x = ± 4 - 2
The solutions are:
p = 4 - 2 = 2
q = - 4 - 2 = - 6
f(x) = ( - 1 ) ( x - p ) ( x - q )
f(x) = - ( x - 2 ) [ ( x - ( - 6 ) ]
f(x) = - ( x - 2 ) ( x + 6 )
To express the given equation, f(x) = -x-4x+ 12, in vertex form, we need to complete the square. The vertex form of a quadratic equation is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Step 1: Combine like terms in the equation.
f(x) = -5x + 12
Step 2: Rearrange the equation to have x terms next to each other.
f(x) = (-5x) + 12
Step 3: Factor out the coefficient of x from the x terms.
f(x) = -1(5x) + 12
Step 4: Find the coefficient (a) by multiplying the coefficient of x by 1/2 and then squaring it.
a = (1/2 * -1)^2 = (1/2)^2 = 1/4
Step 5: Rewrite the equation by adding and subtracting the value found in Step 4 inside the parentheses.
f(x) = -1(5x + 0) + 12 = -1(5x + 0 - 0) + 12
Step 6: Rewrite the equation using the factored form of the quadratic expression.
f(x) = -1(5x + 0 - 0) + 12
= -1(5x - (0)) + 12
Step 7: Rewrite the equation as a perfect square trinomial by squaring the binomial expression inside the parentheses.
f(x) = -1(5x - 0)^2 + 12
Therefore, the equation f(x) = -x - 4x + 12 in vertex form is f(x) = -(5x - 0)^2 + 12.
To express the equation in intercept form, f(x) = a(x + p)(x + q), we need to factorize it using the quadratic formula or by factoring.
Step 1: Rearrange the equation to match the standard form of a quadratic equation (ax^2 + bx + c = 0).
f(x) = 0x^2 - 5x + 12
Step 2: Factorize the quadratic expression by breaking the -5x term into two terms whose coefficients multiply to get the product of the constant term (12) and add up to the coefficient of the x term (-5x).
f(x) = (0x - 3)(0x - 4)
Therefore, the equation f(x) = -x - 4x + 12 in intercept form is f(x) = (0x - 3)(0x - 4).
Now, let's sketch the graph:
The vertex form equation, f(x) = -(5x - 0)^2 + 12, tells us that the vertex of the parabola is located at (0, 12). Since the coefficient a = -1, the parabola opens downwards.
The intercept form equation, f(x) = (0x - 3)(0x - 4), indicates that the x-intercepts occur at x = 3 and x = 4.
Using this information, we can plot the vertex at (0, 12) and the x-intercepts at (3, 0) and (4, 0) on a coordinate plane. Connect these points to form a downward-opening parabolic curve.
The sketch should represent a parabola with its vertex at (0, 12), opening downwards, and x-intercepts at (3, 0) and (4, 0).