Quadratic Graphs And Equations

Quadratic functions have a line of symmetry because they are symmetric about the vertical line passing through the vertex of the parabola. This means that if a point (x, y) lies on the graph of the quadratic function, then the point (-x, y) will also lie on the graph. This line of symmetry divides the parabola into two equal halves.

A quadratic function is a polynomial function of degree 2. Its graph is a parabola, which is a U-shaped curve. The shape of the parabola can vary depending on the coefficients of the quadratic function, but it always has a symmetric form. The graph of a quadratic function can open upwards or downwards, depending on the leading coefficient. Therefore, the correct answer is a parabola.

The graph of a quadratic function can never have three x-intercepts because a quadratic function is a polynomial of degree 2, which means it can have at most two x-intercepts. This is because a quadratic equation can be factored into two linear factors, resulting in two solutions for x. Therefore, the correct answer is no.

The graph of y=x^2 is a parabola that opens upwards. Since the coefficient of the x^2 term is positive, the parabola is concave up and has a minimum point. This minimum point represents the lowest value of the function and is located at the vertex of the parabola. Therefore, the correct answer is minimum.

The equation is a quadratic equation in the form of ax^2 + bx + c = 0. By factoring or using the quadratic formula, we can find the values of x that satisfy the equation. In this case, the equation can be factored as (x + 8)(x + 4) = 0, which gives us x = -8 and x = -4 as the solutions.

The given equation is a quadratic equation in the form of ax^2 + bx + c = 0. To solve this equation, we can use the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a). In this case, a = 8, b = 10, and c = 3. Plugging these values into the quadratic formula, we get x = (-10 ± √(10^2 - 4(8)(3)))/(2(8)). Simplifying further, we get x = (-10 ± √(100 - 96))/(16). This simplifies to x = (-10 ± √4)/(16). Taking the square root, we get x = (-10 ± 2)/(16). Simplifying further, we get x = (-10 + 2)/(16) and x = (-10 - 2)/(16), which gives us x = -0.75 and x = -0.5. Therefore, the answer is x = -0.75 and -0.5.

The given equation is a quadratic equation in the form of ax^2 + bx + c = 0. To solve this equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 16, b = 20, and c = 0. Plugging these values into the quadratic formula, we get x = (-20 ± √(20^2 - 4(16)(0))) / (2(16)). Simplifying further, we have x = (-20 ± √(400)) / 32. This gives us two possible solutions: x = (-20 + 20) / 32 = 0 and x = (-20 - 20) / 32 = -1.25. Therefore, the correct answer is x = 0 and -1.25.

The given equation is a quadratic equation in the form of x^2 - 100 = 0. To solve this equation, we can factorize it as (x - 10)(x + 10) = 0. This means that either (x - 10) = 0 or (x + 10) = 0. Solving these two equations separately, we find that x = 10 and x = -10. Therefore, the correct answer is x = 10 and -10.

The given quadratic equation is in the form ax^2 + bx + c = 0. The discriminant, denoted by Δ, is calculated as b^2 - 4ac. If Δ is greater than 0, the equation has two distinct real solutions. If Δ is equal to 0, the equation has one real solution. If Δ is less than 0, the equation has no real solutions. In this case, the discriminant is 5^2 - 4(2)(7) = 25 - 56 = -31. Since the discriminant is negative, there are no real solutions to the equation.

When we subtract 7 from the equation y = x^2 + 2x, it means that we are subtracting 7 from the y-coordinate of each point on the graph. This will result in the entire graph being shifted downwards by 7 units. Therefore, the correct answer is Down 7.

To change the concavity of the graph of a quadratic function written in standard form, you need to change the value of a to its opposite. The coefficient of the x^2 term, represented by a, determines whether the graph opens upwards or downwards. If a is positive, the graph opens upwards and has a concave shape. If a is negative, the graph opens downwards and has a concave shape. Thus, changing the value of a to its opposite will change the concavity of the graph.

The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)), where f(x) represents the function. In this case, the quadratic function is y = 2x^2 + 12x + 3. By comparing the equation to the standard form, we can see that a = 2, b = 12, and c = 3. Plugging these values into the formula, we get (-12/(2*2), f(-12/(2*2))) = (-3, f(-3)). To find f(-3), we substitute x = -3 into the equation: f(-3) = 2(-3)^2 + 12(-3) + 3 = -15. Therefore, the vertex is (-3, -15).

The given equation is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. In this case, the equation does not factor easily, so we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by x = (-b ± √(b^2 - 4ac)) / (2a). In this equation, a = 1, b = -3, and c = -2. Plugging in these values into the quadratic formula, we get x = (-(-3) ± √((-3)^2 - 4(1)(-2))) / (2(1)). Simplifying further, x = (3 ± √(9 + 8)) / 2, which becomes x = (3 ± √17) / 2. Therefore, the solutions for x are approximately 3.56 and -0.56.

By changing the value of "a" to 3 in the equation y=x^2+4x+7, we are increasing the coefficient of the x^2 term. This will cause the parabola to become steeper and narrower, resulting in a faster growth rate. Increasing the value of "a" will make the graph more concentrated around the vertex, making it grow faster.

The answer (-6,0) and (4,0) suggests that the roots of the equation y = x^2 + 2x - 24 are x = -6 and x = 4. This means that when x is equal to -6 or 4, the equation will be satisfied and y will be equal to 0. Therefore, the points (-6,0) and (4,0) lie on the graph of the equation.