1)Identify the y-intercept and the axis of symemetry for the graph of f(x)=-3x^2+6x+12.
answer=12:x=1
2)Determine whether f(x)=-5x^2-10x+6 has a maximum or minimum value and find that value.
answer=maximum -1
3)Solve x^2=4x by graphing and indicate the roots.
answer=0,4
4)Solve x^2+2x=-2 by graphing and indicate the roots.
answer=-1,1
5)Solve by factoring:x^2-3x=28
answer=-4,7
#1, I got x=-1
#2, no
the function opens up, so there is a minimum.
it occurs when x=-1, and that minimum is f(-1), which is 11
#3, ok
#4 graphing f(x) = x^2 + 2x + 2
produces a parabola which lies above the x-axis, so there is no real solution.
#5 ok
1) To identify the y-intercept, you can set x = 0 and evaluate f(x). In this case, f(x) = -3(0)^2 + 6(0) + 12 = 12. Therefore, the y-intercept is 12.
To find the axis of symmetry, you can use the formula x = -b/2a, where a, b, and c are the coefficients in the quadratic equation f(x) = ax^2 + bx + c. In this case, a = -3 and b = 6. Substituting these values into the formula, we get x = -6/(2*(-3)) = 1. Therefore, the axis of symmetry is x = 1.
2) To determine whether f(x) = -5x^2 - 10x + 6 has a maximum or minimum value, you can look at the coefficient of x^2. If this coefficient is positive, then the parabola opens upwards and has a minimum value. If it is negative, then the parabola opens downwards and has a maximum value. In this case, the coefficient is -5, so the parabola opens downwards and has a maximum value.
To find the value, you can use the formula x = -b/2a to determine the x-coordinate of the vertex (where the maximum/minimum occurs). In this case, a = -5 and b = -10. Substituting these values into the formula, we get x = -(-10)/(2*(-5)) = -1. Plug in this value of x into the equation to find the corresponding y-coordinate: f(-1) = -5(-1)^2 - 10(-1) + 6 = -5 + 10 + 6 = 11. Therefore, the maximum value is 11.
3) To solve x^2 = 4x by graphing, you can plot the graph of both sides of the equation. The roots of the equation (where the graph intersects the x-axis) will be the values of x where the graph crosses the x-axis. In this case, the left side of the equation is a parabola that opens upwards, while the right side is a linear equation that represents a straight line. They intersect at points (0, 0) and (4, 4) on the graph. Therefore, the roots of the equation are 0 and 4.
4) To solve x^2 + 2x = -2 by graphing, similarly to the previous question, you can plot the graph of both sides of the equation. The roots of the equation will be the x-values where the graph intersects the x-axis. In this case, the left side of the equation is a parabola that opens upwards while the right side is a constant (-2). Because the parabola lies entirely above the x-axis and does not intersect it, there are no real solutions to this equation.
5) To solve x^2 - 3x = 28 by factoring, you can rearrange the equation to x^2 - 3x - 28 = 0. Then, you need to find two numbers whose product is -28 and whose sum is -3 (because the coefficient of x is -3). These numbers are -4 and 7. Therefore, you can factor the equation as (x - 7)(x + 4) = 0. Set each factor equal to zero and solve for x to get x = -4 and x = 7. Therefore, the roots of the equation are -4 and 7.