Question

warm up 35
identify the x-intercepts,
y-intercepts, vertex,
maximum/minimum, axis
of symmetry.

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph intersects the x - axis (where \(y = 0\)). From the graph, we can see that the parabola intersects the x - axis at \(x=-2\) and \(x = 2\). So the x - intercepts are \((-2,0)\) and \((2,0)\).

Step2: Identify y - intercept

The y - intercept is the point where the graph intersects the y - axis (where \(x = 0\)). Looking at the graph, when \(x = 0\), \(y=-4\). So the y - intercept is \((0,-4)\).

Step3: Identify vertex

The vertex of a parabola is the lowest (for a parabola opening upwards) or highest (for a parabola opening downwards) point. Since the parabola opens upwards, the vertex is the minimum point. From the graph, the vertex is at \((0,-4)\).

Step4: Identify maximum/minimum

Since the parabola opens upwards (the coefficient of \(x^{2}\) is positive, as we can infer from the shape), the parabola has a minimum value. The minimum value occurs at the vertex, and the minimum value of \(y\) is \(-4\) (at \(x = 0\)).

Step5: Identify axis of symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex. For a parabola of the form \(y = ax^{2}+bx + c\), the axis of symmetry is given by \(x=-\frac{b}{2a}\). For this parabola, since the vertex is at \((0,-4)\), the axis of symmetry is the vertical line \(x = 0\) (the y - axis).

Answer:

• x - intercepts: \((-2,0)\), \((2,0)\)
• y - intercept: \((0,-4)\)
• vertex: \((0,-4)\)
• minimum: \(y=-4\) (at \(x = 0\)), no maximum (since the parabola opens upwards and extends to \(+\infty\))
• axis of symmetry: \(x = 0\) (the y - axis)