To determine which graph corresponds to the function $y = x^4 - 5x^2 + 4$, let's first analyze its general shape and key features.

1. General Shape (Even degree polynomial):

The highest degree of the function is 4, which means the graph will have a general "U" shape. Specifically:

• As $x \to \infty$, $y \to \infty$ (since $x^4$ dominates at large values of $x$).
• As $x \to -\infty$, $y \to \infty$ (since $x^4$ also dominates at large negative values of $x$).

Thus, the ends of the graph both point upwards.

2. Symmetry:

Since the function only has even powers of $x$ (no odd terms), the graph will be symmetric about the y-axis. This is because for every positive $x$, $f(x) = f(-x)$.

3. Intercepts:

• Y-intercept: Set $x = 0$. $y = 0^4 - 5(0)^2 + 4 = 4$ So, the y-intercept is at $(0, 4)$.

• X-intercepts (roots): Set $y = 0$ and solve the equation $x^4 - 5x^2 + 4 = 0$. Let $z = x^2$, then the equation becomes: $z^2 - 5z + 4 = 0$ Solving this quadratic equation: $z = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2}$ So, $z = 4$ or $z = 1$.

  Since $z = x^2$, we have two cases:

  • $x^2 = 4$, which gives $x = \pm 2$,
  • $x^2 = 1$, which gives $x = \pm 1$.

  Thus, the x-intercepts are $x = -2, -1, 1, 2$, and the corresponding points are $(-2, 0), (-1, 0), (1, 0), (2, 0)$.

4. Turning Points:

To find the critical points, we differentiate $y = x^4 - 5x^2 + 4$: $\frac{dy}{dx} = 4x^3 - 10x$ Setting this equal to 0 to find the critical points: $4x(x^2 - 2.5) = 0$ Thus, $x = 0$ or $x = \pm \sqrt{2.5}$.

• $x = 0$ is a critical point, and solving for $\pm \sqrt{2.5}$ will give us the x-coordinates of the other turning points.

Conclusion:

The graph is symmetric about the y-axis, has a minimum at $x = 0$, crosses the x-axis at $x = -2, -1, 1, 2$, and extends upwards at the far left and right.

Given this analysis, you can compare these features with the options for the graph of the function. Does this make sense, and do you want more details about the graph's turning points?

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Here are some related questions:

1. How would you solve for turning points of a higher degree polynomial?
2. Can you explain how the symmetry of even-powered functions affects their graph?
3. What happens to the shape of a polynomial graph when higher-order terms dominate?
4. Why does the function have 4 x-intercepts, and how does that relate to its degree?
5. What role do y-intercepts play in determining a graph’s key features?

Tip: To sketch a polynomial function, always start with finding intercepts and considering the behavior at extremes (large positive and negative x-values).