How do I find the vertex, axis of symmetry, y-intercept, x-intercept, domain and range of #y=-x^2+2x-5#?

Answer 1

y = -x^2 + 2x - 5

y = -x^2 + 2x - 5 x-coordinate of vertex and axis of symmetry: x = (-b/2a) = -2/-2 = 1 y-coordinate of vertex: y(1) = -1 + 2 - 5 = -4 y-intercept. Make x = 0 --> y = -5 x-intercepts. Make y = 0 -->Solve -x^2 + 2x - 5 = 0. Since D = 4 - 20 < 0, there are no real roots (no x-intercepts) The graph is a parabola that opens downward (a < 0), and that is completely below the x-axis. It has a minimum. (-4) at vertex (1, -4). Domain of x (-infinity, infinity) Range of y (-infinity, -4)and (-4, -infinity) graph{-x^2 + 2x - 5 [-20, 20, -10, 10]}

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Answer 2
To find the vertex, axis of symmetry, y-intercept, x-intercept, domain, and range of the given quadratic function \( y = -x^2 + 2x - 5 \), follow these steps: 1. **Vertex:** The vertex of a quadratic function \( y = ax^2 + bx + c \) is given by the formula \( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \). 2. **Axis of Symmetry:** The axis of symmetry of a quadratic function is the vertical line passing through the vertex. It is given by the equation \( x = -\frac{b}{2a} \). 3. **Y-Intercept:** To find the y-intercept, set \( x = 0 \) and solve for y. The y-intercept is the point (0, y). 4. **X-Intercept:** To find the x-intercepts, set \( y = 0 \) and solve for x. The x-intercepts are the points where the graph intersects the x-axis. 5. **Domain:** The domain of a quadratic function is all real numbers unless there are restrictions due to square roots or denominators. 6. **Range:** The range of a quadratic function in the form \( y = ax^2 + bx + c \) is \( (-\infty, c] \) if \( a < 0 \) and \( [c, \infty) \) if \( a > 0 \). For the given function \( y = -x^2 + 2x - 5 \): 1. **Vertex:** \( \left( 1, -6 \right) \) 2. **Axis of Symmetry:** \( x = 1 \) 3. **Y-Intercept:** \( (0, -5) \) 4. **X-Intercepts:** None (the parabola does not intersect the x-axis) 5. **Domain:** All real numbers 6. **Range:** \( (-\infty, -5] \)
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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