Graph the quadratic function f(x) = 2x² - 8x + 6, identify its vertex, axis of symmetry, and x-intercepts.

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Let's analyze the quadratic function f(x) equals 2x squared minus 8x plus 6. We'll identify its key features including the vertex, axis of symmetry, and x-intercepts. First, we need to graph the function. The function is in the form ax squared plus bx plus c, where a equals 2, b equals negative 8, and c equals 6. Since a is positive, the parabola opens upward. To find the vertex, we use the formula negative b divided by 2a for the x-coordinate. This gives us 8 divided by 4, which equals 2. The y-coordinate is found by evaluating f of 2, which equals negative 2. So the vertex is at the point (2, negative 2). The axis of symmetry passes through the vertex, so its equation is x equals 2. Now, let's find the x-intercepts of our quadratic function. These are the points where the graph crosses the x-axis, which means f(x) equals zero. To find these points, we set the equation equal to zero: 2x squared minus 8x plus 6 equals 0. First, we divide everything by 2 to simplify: x squared minus 4x plus 3 equals 0. Next, we factor this expression: (x minus 1) times (x minus 3) equals 0. Using the zero product property, either x minus 1 equals 0, or x minus 3 equals 0. Solving these equations gives us x equals 1 or x equals 3. So our x-intercepts are at the points (1, 0) and (3, 0). Notice how these points are symmetrically placed on either side of the axis of symmetry at x equals 2. Let's complete our analysis of the quadratic function f(x) equals 2x squared minus 8x plus 6. We've already identified the vertex at (2, negative 2), the axis of symmetry at x equals 2, and the x-intercepts at (1, 0) and (3, 0). Now, let's find the y-intercept, which occurs when x equals 0. Substituting x equals 0 into our function, we get f(0) equals 2 times 0 squared minus 8 times 0 plus 6, which simplifies to 6. So the y-intercept is at the point (0, 6). We can also express our function in vertex form as f(x) equals 2 times (x minus 2) squared minus 2. This form clearly shows that the vertex is at (2, negative 2) and the parabola opens upward with a leading coefficient of 2. Notice how all these key features are visible on our graph: the vertex at the lowest point, the axis of symmetry passing through the vertex, and the x and y-intercepts where the graph crosses the axes. Now let's explore how the parameters in a quadratic function affect its graph. The standard form of a quadratic function is f(x) equals a x-squared plus b x plus c. We can also write it in vertex form as f(x) equals a times (x minus h) squared plus k, where (h,k) represents the vertex. The parameter 'a' determines both the direction and width of the parabola. When a is positive, like in our function where a equals 2, the parabola opens upward. If a were negative, the parabola would open downward, as shown by the red curve. The absolute value of 'a' controls the width - a larger absolute value creates a narrower parabola, as demonstrated by the green curve where a equals 4. The parameters h and k determine the position of the vertex. In our function, h equals 2 and k equals negative 2, which means the vertex is at the point (2, negative 2). Understanding these transformations helps us quickly sketch and analyze any quadratic function. Let's summarize what we've learned about the quadratic function f(x) equals 2x squared minus 8x plus 6. We identified that this parabola has its vertex at the point (2, negative 2), which is the minimum point of the function since the parabola opens upward. The axis of symmetry is the vertical line x equals 2, which passes through the vertex and divides the parabola into two mirror-image halves. We found that the x-intercepts are at the points (1, 0) and (3, 0), where the graph crosses the x-axis and f(x) equals zero. The y-intercept is at the point (0, 6), where the graph crosses the y-axis. We also expressed the function in vertex form as f(x) equals 2 times (x minus 2) squared minus 2, which clearly shows that the leading coefficient a equals 2, the horizontal shift h equals 2, and the vertical shift k equals negative 2. These key features provide a complete understanding of the quadratic function's behavior and graph.