HELP Not great in these
Express y=2x^2 -12x +23 in the form
y=2(x-c)^2 + d
The graph of y=x^2 is transformed into the graph of y=2x^2 - 12x +23 by the transformation
a vertical stretch with scale factor k followed by,
A horizontal translation of p units followed by,
a vertical translation of q units
what are the values of k, p , and q
HELP PLEASE!
2x^2 -12x +23=2*(x^2-6x+23/2)
(x-a)^2= x^2-2ax+a^2
(x-3)^2= x^2-2*3*x+3^2= x^2-6x+3^2
2*(x^2-6x+23/2)=
=2*(x^2-2*3*x +3^2+ 23/2 -3^2)
=2(x^2-6x+3^2)+2*(23/2)-2*(3^2)
=2*(x-3)^2+23-2*9
=2*(x-3)^2+23-18
=2*(x-3)^2 +5
Proof:
2*(x-3)^2+5=2*(x^2-2*3*x+3^2)+5
=2*(x^2-6x+9)+5
=2x^2-12x+18+5=2x^2-12x+23
So:
2x^2 -12x +23 = 2*(x-3)^2 +5
a vertical stretch =2
a y-intercept is a point where the graph of a function intersects with the y-axis
2*0^2-12*0+23=0-0+23=23
vertical translation = 23
Quadratic function have extreme point
for x=-(b/2a)
In this case a=2 b=-12
-(b/2a)=-[(-12/(2*2]=-(-12/4)=12/4=3
2x^2 -12x +23 = 2*(x-3)^2 +5
k=2
p=3
q=23
Oh boy, it's math time! Don't worry, I'm here to help! Let's transform this equation step by step.
First, we want to express the equation y = 2x^2 - 12x + 23 in the form y = 2(x - c)^2 + d. This means we need to complete the square.
To do that, we can focus on the x^2 and x terms.
Let's start with the x^2 term. In the form y = a(x - h)^2 + k, we have "h" as the x-coordinate of the vertex. In this case, h would be the x-coordinate caused by completing the square.
The formula for completing the square is (x - (b/2a))^2, where b is the coefficient of the x term. In our case, b = -12, and a = 2 since the coefficient of x^2 is 2.
Using the completing the square formula, we get:
x - (-12/2*2))^2 = (x - (-12/4))^2 = (x + 3)^2.
Now let's focus on the x term. Since we already handled the x^2 term, we don't need to touch it.
The final part of the equation is the constant term, 23. And that will be equivalent to "d" in the form y = 2(x - c)^2 + d.
Putting it all together, we have:
y = 2(x + 3)^2 + 23.
Now, let's move on to the second part of your question. You mentioned that the graph of y = x^2 was transformed into y = 2x^2 - 12x + 23.
From the given information, we can deduce that the transformation involves:
- A vertical stretch with a scale factor of k: Since the coefficient of x^2 changed from 1 to 2, we can tell that there was a vertical stretch by a factor of 2.
- A horizontal translation of p units: This is indicated by the (x + 3) term, which moves the graph 3 units to the left.
- A vertical translation of q units: Since there's no explicit term for a vertical translation, we can see that the constant term, 23, acts as the vertical translation value. So the graph was shifted up by 23 units.
So, to summarize:
- The scale factor, k, is 2 (indicating a vertical stretch).
- The horizontal translation, p, is -3 (indicating a shift 3 units to the left).
- The vertical translation, q, is 23 (indicating a shift 23 units up).
I hope that clears things up! If you have any more questions, feel free to ask!
To express y = 2x^2 - 12x + 23 in the form y = 2(x - c)^2 + d, we can complete the square.
First, we factor out the common factor of 2 from the terms involving x:
y = 2(x^2 - 6x) + 23
To complete the square, we need to take half of the coefficient of x, square it, and add it both inside and outside the parentheses:
y = 2(x^2 - 6x + (-6/2)^2) + 23 - 2((-6/2)^2)
Simplifying further:
y = 2(x^2 - 6x + 9) + 23 - 2(9)
y = 2(x - 3)^2 + 23 - 18
y = 2(x - 3)^2 + 5
Comparing this to the desired form, we can see that c = 3 and d = 5.
Now, let's determine the values of k, p, and q for the transformation of the graph of y = x^2 to y = 2x^2 - 12x + 23.
The vertical stretch with a scale factor k can be found by comparing the coefficient of x^2 in y = 2x^2 - 12x + 23 to y = x^2. Here, k = 2, which means the graph is vertically stretched by a factor of 2.
The horizontal translation by p units can be determined by equating the linear terms -12x in y = 2x^2 - 12x + 23 with the linear term -2px in y = 2(x - p)^2 + 5. Simplifying this equation, we have -12x = -2px, which gives p = 6. Therefore, the graph is horizontally translated 6 units to the right.
The vertical translation by q units can be found by comparing the constant term 23 in y = 2x^2 - 12x + 23 to the constant term d in y = 2(x - c)^2 + d. Here, q = 5, which means the graph is vertically shifted 5 units upwards.
To summarize, the transformation from the graph of y = x^2 to y = 2x^2 - 12x + 23 can be described as:
- Vertical stretch with a scale factor of 2
- Horizontal translation of 6 units to the right
- Vertical translation of 5 units upwards.
To express the equation y=2x^2-12x+23 in the form y=2(x-c)^2+d, you need to complete the square. Follow these steps to find the values of c and d:
Step 1: Divide the coefficient of x by 2 and square it:
c = (-12 / 2)^2 = (-6)^2 = 36
Step 2: Substitute the value of c into the original equation and simplify:
y = 2x^2 - 12x + 23
y = 2(x^2 - 6x + 36) + 23 - (2 * 36)
y = 2(x - 6)^2 + 23 - 72
y = 2(x - 6)^2 - 49
Now, let's determine the values of k, p, and q by analyzing the given transformation:
1. Vertical Stretch (Scale factor k):
Since the original equation is y = x^2, and the transformed equation is y = 2x^2 - 12x + 23, we can see that the transformed equation has been stretched vertically by a factor of 2. Therefore, k = 2.
2. Horizontal Translation (p units):
To determine the horizontal translation, we need to find the value of c, which is the x-coordinate of the vertex. In this case, c is 6. Since the original equation is y = x^2 and the transformed equation is y = 2(x - 6)^2 - 49, we can see that the graph has been shifted horizontally 6 units to the right. Therefore, p = 6.
3. Vertical Translation (q units):
By comparing the original equation y = x^2 to the transformed equation y = 2(x - 6)^2 - 49, we can see that the graph has been shifted vertically downwards by 49 units. Therefore, q = -49.
To summarize:
k = 2
p = 6
q = -49