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Math  /  Algebra

QuestionGraph each equation. 5) y=x2−2x−3y=x^{2}-2 x-3
Identify the min⁡/max⁡\min / \max value of each. Th

Studdy Solution

STEP 1

1. The function y=x2−2x−3 y = x^2 - 2x - 3 is a quadratic equation.
2. The graph of a quadratic equation is a parabola.
3. The parabola opens upwards if the coefficient of x2 x^2 is positive, and downwards if it is negative.
4. The vertex of the parabola represents the minimum or maximum value of the function.

STEP 2

1. Identify the direction of the parabola.
2. Find the vertex of the parabola.
3. Determine if the vertex is a minimum or maximum.
4. Graph the parabola.
5. Identify the minimum or maximum value from the vertex.

STEP 3

Identify the direction of the parabola by examining the coefficient of x2 x^2 .
The equation is y=x2−2x−3 y = x^2 - 2x - 3 . The coefficient of x2 x^2 is 1, which is positive.
Thus, the parabola opens upwards.

STEP 4

Find the vertex of the parabola using the vertex formula x=−b2a x = -\frac{b}{2a} , where a a and b b are the coefficients from the quadratic equation ax2+bx+c ax^2 + bx + c .
Here, a=1 a = 1 and b=−2 b = -2 .
x=−−22×1=22=1 x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1
Now, substitute x=1 x = 1 back into the equation to find the y y -coordinate of the vertex:
y=(1)2−2(1)−3=1−2−3=−4 y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4
The vertex is at (1,−4) (1, -4) .

STEP 5

Determine if the vertex is a minimum or maximum.
Since the parabola opens upwards, the vertex represents the minimum point of the parabola.

STEP 6

Graph the parabola.
1. Plot the vertex at (1,−4) (1, -4) .
2. Identify the axis of symmetry, which is the vertical line x=1 x = 1 .
3. Choose additional points on either side of the vertex to plot, such as x=0 x = 0 and x=2 x = 2 .
For x=0 x = 0 :
y=(0)2−2(0)−3=−3 y = (0)^2 - 2(0) - 3 = -3 Point: (0,−3) (0, -3)
For x=2 x = 2 :
y=(2)2−2(2)−3=4−4−3=−3 y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3 Point: (2,−3) (2, -3)
4. Plot these points and draw a smooth curve through them to complete the parabola.

STEP 7

Identify the minimum value from the vertex.
The minimum value of the function is the y y -coordinate of the vertex, which is −4 -4 .
The minimum value of the function y=x2−2x−3 y = x^2 - 2x - 3 is −4 \boxed{-4} .

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