Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=x²+3x−10

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = x^2 + 3x - 10\).
2. Find the vertex of the parabola using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = 1\) and \(b = 3\).
3. Calculate the y-coordinate of the vertex by substituting the x-value found into the function: \(f\left(-\frac{b}{2a}\right)\).
4. Find the x-intercepts by solving the quadratic equation \(x^2 + 3x - 10 = 0\) using factoring or the quadratic formula.
5. Determine the y-intercept by evaluating \(f(0)\), then write the equation of the axis of symmetry as \(x = -\frac{b}{2a}\), and use the vertex and intercepts to describe the domain and range of the function.

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. It can be found using the formula x = -b/(2a) for a quadratic function f(x) = ax² + bx + c. The vertex helps in sketching the graph and determining the axis of symmetry.

Intercepts of a Quadratic Function

Intercepts are points where the graph crosses the axes. The y-intercept is found by evaluating f(0), and the x-intercepts (roots) are found by solving f(x) = 0. These points provide key reference locations for sketching the parabola accurately.

Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward, the range is all values greater than or equal to the vertex's y-coordinate; if downward, all values less than or equal to it. This helps describe the function's output values.