Question

key features of quadratic functions
label the x-intercept(s), y-intercept, vertex, and line of symmetry of the following graph.

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph intersects the x - axis (where \(y = 0\)). For the left parabola, looking at the graph, the points where \(y = 0\) are at \(x=-3\) (approximate, from the graph's intersection with the x - axis) and \(x = 1\) (approximate). Wait, actually, from the grid, the left parabola crosses the x - axis at \(x=-3\) (let's check the grid: the x - axis is horizontal, the points on the x - axis for the left parabola are at \(x=-3\) (between - 5 and 0) and \(x = 1\) (between 0 and 5)? Wait, no, looking at the left graph: the x - intercepts are the two points where the parabola crosses the x - axis. From the graph, one is at \(x=-3\) (since it's between - 5 and 0, closer to - 3? Wait, the grid lines: the x - axis has marks at - 5, 0, 5. The left parabola crosses the x - axis at \(x=-3\) (let's say) and \(x = 1\)? Wait, no, maybe the x - intercepts are at \(x=-3\) and \(x = 1\) (but actually, looking at the left graph, the two points on the x - axis are at \(x=-3\) (approx) and \(x = 1\) (approx). Wait, maybe more accurately, from the graph, the left parabola (opening upwards) has x - intercepts at \(x=-3\) and \(x = 1\) (the points where \(y = 0\)).

Step2: Identify y - intercept

The y - intercept is the point where the graph intersects the y - axis (where \(x = 0\)). For the left parabola, when \(x = 0\), the y - coordinate is - 4 (approx, from the graph, the point on the y - axis is at \(y=-4\) or so, but looking at the grid, the vertex is at \((-1,-4)\) maybe? Wait, no, the vertex of the left parabola (opening upwards) is the lowest point. The vertex is at \((-1,-4)\) (approx, since the line of symmetry is \(x=-1\), as the parabola is symmetric about \(x=-1\)). The y - intercept is the point where \(x = 0\), so for the left parabola, when \(x = 0\), the y - value is - 3? Wait, maybe better to look at the left graph: the y - intercept is the point on the y - axis (x = 0). From the graph, the left parabola intersects the y - axis at \(y=-3\) (approx).

Step3: Identify vertex

The vertex of a parabola is the minimum (for opening upwards) or maximum (for opening downwards) point. For the left parabola (opening upwards), the vertex is the lowest point. From the graph, the vertex is at \((-1,-4)\) (since the line of symmetry is \(x=-1\), and the y - coordinate at \(x=-1\) is the minimum).

Step4: Identify line of symmetry

The line of symmetry of a parabola is a vertical line that passes through the vertex. For the left parabola, since the vertex is at \(x=-1\), the line of symmetry is \(x=-1\).

For the right parabola (opening downwards):

Step1: Identify x - intercepts

The right parabola (opening downwards) intersects the x - axis at \(x = 1\) and \(x = 4\) (approx, from the graph, where \(y = 0\)).

Step2: Identify y - intercept

The y - intercept is where \(x = 0\). For the right parabola, when \(x = 0\), the y - coordinate is 4 (approx, from the graph, the point on the y - axis is at \(y = 4\)).

Step3: Identify vertex

The vertex of the right parabola (opening downwards) is the highest point. From the graph, the vertex is at \((2,9)\) (approx, since the line of symmetry is \(x = 2\), and the y - coordinate at \(x = 2\) is the maximum).

Step4: Identify line of symmetry

The line of symmetry for the right parabola is \(x = 2\), since it passes through the vertex.

But maybe more precisely, for the left parabola (opening upwards):

- x - intercepts: \((-3,0)\) and \((1,0)\)
- y - intercept: \((0,-3)\)
- vertex: \((-1,-4)\)
- line of symmetry: \(x=-1\)

For the right parabola (opening downwards):

- x - intercepts: \((1,0)\) and \((4,0)\) (wait, no, looking at the right graph, it crosses the x - axis at \(x = 1\) and \(x = 4\) (approx))
- y - intercept: \((0,4)\) (approx)
- vertex: \((2,9)\) (approx)
- line of symmetry: \(x = 2\)

Answer:

For the left (upward - opening) parabola:

• x - intercepts: \((-3,0)\) and \((1,0)\)
• y - intercept: \((0,-3)\)
• vertex: \((-1,-4)\)
• line of symmetry: \(x=-1\)

For the right (downward - opening) parabola:

• x - intercepts: \((1,0)\) and \((4,0)\) (approx)
• y - intercept: \((0,4)\) (approx)
• vertex: \((2,9)\) (approx)
• line of symmetry: \(x = 2\)

(Note: The exact values may vary slightly based on the graph's scale, but these are the approximate key features.)